Discrete

📁 Source: Mathlib/CategoryTheory/Monoidal/Discrete.lean

Statistics

Metric	Count
Definitions Discrete, addMonoidal, addMonoidalFunctor, addMonoidalFunctorComp, addMonoidalFunctorMonoidal, monoidal, monoidalFunctor, monoidalFunctorComp, monoidalFunctorMonoidal	9
Theorems addMonoidalFunctorComp_isMonoidal, addMonoidalFunctor_obj, addMonoidalFunctor_δ, addMonoidalFunctor_ε, addMonoidalFunctor_η, addMonoidalFunctor_μ, addMonoidal_associator, addMonoidal_leftUnitor, addMonoidal_rightUnitor, addMonoidal_tensorObj_as, addMonoidal_tensorUnit_as, monoidalFunctorComp_isMonoidal, monoidalFunctor_obj, monoidalFunctor_δ, monoidalFunctor_ε, monoidalFunctor_η, monoidalFunctor_μ, monoidal_associator, monoidal_leftUnitor, monoidal_rightUnitor, monoidal_tensorObj_as, monoidal_tensorUnit_as	22
Total	31

CategoryTheory

Definitions

Name	Category	Theorems
Discrete 📖	CompData	2375 math math: TopCat.Presheaf.generateEquivalenceOpensLe_functor'_obj_obj, TopCat.isInducing_pullback_to_prod, CommRingCat.tensorProd_map_right, FinitaryPreExtensive.isIso_sigmaDesc_fst, Discrete.monoidal_associator, IsGrothendieckAbelian.GabrielPopescuAux.ι_d, Over.associator_hom_left_snd_fst_assoc, CostructuredArrow.homMk'_id, CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_obj, TopCat.binaryCofan_isColimit_iff, ChosenPullbacksAlong.Over.whiskerLeft_left_fst, Functor.LeftExtension.coconeAtFunctor_map_hom, SimplicialObject.Splitting.cofan_inj_πSummand_eq_id_assoc, TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_hom_right, Over.prodLeftIsoPullback_hom_snd_assoc, LocalizerMorphism.IsRightDerivabilityStructure.Constructor.fromRightResolution_map, PreOneHypercover.forkOfIsColimit_ι_map_inj_assoc, StructuredArrow.projectSubobject_mk, Over.μ_pullback_left_snd', WithTerminal.coneEquiv_unitIso_hom_app_hom_left, Limits.limitConeOfUnique_cone_π, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc, Bicategory.RightExtension.w_assoc, Limits.BinaryFan.rightUnitor_hom, Discrete.sumEquiv_counitIso_inv_app, CostructuredArrow.ofCommaFstEquivalenceInverse_obj_left_hom, Over.ConstructProducts.has_over_limit_discrete_of_widePullback_limit, StructuredArrow.map_map_right, CommMon.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon_map, StructuredArrow.ofCommaSndEquivalenceInverse_obj_right_left, ChosenPullbacksAlong.Over.associator_hom_left_snd_fst_assoc, AddCommGrpCat.binaryProductLimitCone_cone_pt, Limits.Bicone.π_of_isColimit, Limits.inr_of_isLimit, Condensed.instAB4StarCondensedMod, finitaryExtensive_iff_of_isTerminal, Limits.diagramIsoPair_hom_app, CostructuredArrow.hom_eq_iff, PreservesPullbacksOfInclusions.preservesPullbackInl', StructuredArrow.ofCommaSndEquivalenceFunctor_map_right, Limits.FormalCoproduct.isColimitCofan_desc_φ, AlgebraicGeometry.Scheme.Cover.pullbackCoverOver_X, Limits.Bicone.toCocone_ι_app_mk, Limits.Types.Small.productIso_hom_comp_eval, Over.ConstructProducts.conesEquivInverseObj_pt, extendFan_pt, CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_hom_app, Functor.leftExtensionEquivalenceOfIso₁_functor_map_left, Limits.Types.Small.productIso_hom_comp_eval_apply, Over.iteratedSliceForwardNaturalityIso_hom_app, PreGaloisCategory.FiberFunctor.instReflectsColimitsOfShapeFintypeCatDiscretePEmpty, CostructuredArrow.mk_hom_eq_self, Functor.LeftExtension.precomp₂_obj_hom_app, Limits.Types.binaryCofan_isColimit_iff, CostructuredArrow.toOver_obj_left, StructuredArrow.ofStructuredArrowProjEquivalence.inverse_obj_right_right, MonoOver.mk_coe, ChosenPullbacksAlong.Over.tensorHom_left_fst, TopCat.GlueData.π_surjective, StructuredArrow.map_obj_right, StructuredArrow.mapIso_functor_obj_left, CategoryOfElements.fromStructuredArrow_map, Limits.MulticospanIndex.multiforkEquivPiFork_counitIso_hom_app_hom, Limits.CofanTypes.isColimit_iff_bijective_fromSigma, Limits.coprod.symmetry, instFiniteArrowDiscrete, Limits.MulticospanIndex.multiforkEquivPiFork_unitIso_inv_app_hom, Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_isLimit_lift, ChosenPullbacksAlong.cartesianMonoidalCategoryFst_pullback_map, Bicategory.LeftLift.whiskering_map, PreservesPullbacksOfInclusions.preservesPullbackInr, MonoOver.congr_unitIso, AlgebraicGeometry.coprodSpec_apply, NonPreadditiveAbelian.diag_σ, Mon.EquivLaxMonoidalFunctorPUnit.isMonHom_counitIsoAux, FunctorToTypes.prodMk_fst, Functor.preservesBinaryCoproducts_of_preservesCokernels, OverPresheafAux.unitAux_hom, Over.iteratedSliceBackward_map, Functor.LeftExtension.precomp₂_map_right, CategoryOfElements.costructuredArrowULiftYonedaEquivalence_unitIso, CostructuredArrow.ofDiagEquivalence.functor_map_left_left, monoidalOfHasFiniteCoproducts.whiskerLeft, AlgebraicGeometry.Scheme.Hom.toNormalization_inl_normalizationCoprodIso_hom_assoc, Over.associator_inv_left_snd, Discrete.sumEquiv_unitIso_inv_app, Limits.coprod.associator_naturality, SmallObject.ρFunctorObj_π_assoc, AlgebraicGeometry.instIsIsoSchemeCoprodComparisonOppositeCommRingCatSpec, FreeMonoidalCategory.inclusion_obj, Comma.toPUnitIdEquiv_functor_map, whiskerLeft_coprod_inr_leftDistrib_inv, StructuredArrow.commaMapEquivalenceInverse_map, Limits.limitSubobjectProduct_mono, Under.postComp_inv_app_right, Over.pullback_obj_left, Bicategory.LeftExtension.w_assoc, Limits.instPreservesColimitsOfShapeDiscreteOfFiniteOfPreservesFiniteCoproducts, AlgebraicGeometry.Scheme.Hom.inr_toNormalization_normalizationCoprodIso_inv, Limits.MultispanIndex.inj_sndSigmaMapOfIsColimit, Over.inv_left_hom_left_assoc, Limits.CofanTypes.isColimit_sigma, coprod_inr_rightDistrib_hom_assoc, TopCat.Presheaf.generateEquivalenceOpensLe_unitIso, Limits.Types.pi_map_π_apply, FreeMonoidalCategory.normalizeIsoAux_inv_app, StructuredArrow.ofDiagEquivalence.inverse_map_right, StructuredArrow.w_prod_fst, PreGaloisCategory.FiberFunctor.instReflectsLimitsOfShapeFintypeCatDiscretePEmpty, PreOneHypercover.forkOfIsColimit_pt, TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_hom, Functor.instIsLeftAdjointDiscreteTensorLeftCompIncl, Limits.Pi.cone_π, StructuredArrow.homMk'_comp, Under.forgetMapInitial_inv_app, CostructuredArrow.w_assoc, Comma.toIdPUnitEquiv_inverse_map_right, WithInitial.isColimitEquiv_apply_desc_right, AlgebraicGeometry.instIsAffineSigmaObjScheme, MonoOver.isIso_left_iff_subobjectMk_eq, Mon.equivLaxMonoidalFunctorPUnit_inverse, Limits.coprod.associator_inv, Sieve.overEquiv_pullback, Over.rightUnitor_inv_left_fst_assoc, Limits.BinaryBicone.toCone_π_app_right, Functor.initial_of_isCofiltered_pUnit, MorphismProperty.Over.hasPullbacks, Limits.MulticospanIndex.sndPiMapOfIsLimit_proj, WithInitial.coconeEquiv_functor_obj_pt, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand₀', OverPresheafAux.restrictedYoneda_map, Discrete.opposite_functor_obj_as, toOver_obj_hom, Limits.MultispanIndex.multicoforkEquivSigmaCoforkOfIsColimit_counitIso, Limits.prod.pentagon_assoc, HasPullbacksOfInclusions.preservesPullbackInl', StructuredArrow.ofStructuredArrowProjEquivalence.inverse_obj_right_left_as, Limits.BinaryBicone.ofColimitCocone_inl, Bicategory.RightLift.w_assoc, Sieve.ofArrows_category', Over.comp_left_assoc, Limits.BinaryFan.braiding_hom_snd_assoc, Under.epi_right_of_epi, TopCat.Presheaf.SheafConditionEqualizerProducts.piOpens.hom_ext_iff, Discrete.monoidal_tensorObj_as, Limits.FormalCoproduct.coproductIsoSelf_inv_f, CostructuredArrow.preEquivalence.functor_obj_right_as, MorphismProperty.Over.instHasTerminalTopOfContainsIdentities, Limits.MultispanIndex.multicoforkEquivSigmaCofork_functor_obj_ι_app, PreOneHypercover.p₁_sigmaOfIsColimit_assoc, StructuredArrow.mapIso_inverse_obj_hom, Functor.EssImageSubcategory.associator_inv_def, SmallObject.ι_functorMapSrc, Under.postCongr_inv_app_right, Under.mono_right_of_mono, Limits.MultispanIndex.multicoforkEquivSigmaCofork_counitIso_inv_app_hom, Limits.BinaryBicone.ofLimitCone_inl, CostructuredArrow.pre_obj_hom, Over.hom_left_inv_left, StructuredArrow.toUnder_obj_left, preservesFiniteColimits_liftToFinset, Functor.RightExtension.postcompose₂_obj_left_map, AlgebraicTopology.DoldKan.Γ₀.Obj.map_epi_on_summand_id_assoc, AlgebraicGeometry.sigmaOpenCover_X, Limits.PreservesLimitsOfShape.ofWidePullbacks, TopCat.sigmaIsoSigma_hom_ι, PreZeroHypercover.sigmaOfIsColimit_X, GlueData.types_π_surjective, IsPullback.of_isLimit_binaryFan_of_isTerminal, Limits.MulticospanIndex.multiforkEquivPiFork_functor_map_hom, Limits.FormalCoproduct.cofanHomEquiv_apply_f, Over.whiskerLeft_left, IsSifted.colim_preservesBinaryProducts_of_isSifted, CostructuredArrow.grothendieckPrecompFunctorToComma_map_left, Grpd.piIsoPi_hom_π, Over.forgetMapTerminal_hom_app, OverPresheafAux.restrictedYoneda_obj, Over.mk_left, Bicategory.LeftExtension.ofCompId_right, Presheaf.isSheaf_iff_isLimit_coverage, AlgebraicGeometry.ι_right_coprodIsoSigma_inv, MonoOver.isIso_iff_subobjectMk_eq, Functor.RightExtension.postcompose₂_obj_right, Functor.LeftExtension.postcomp₁_map_right_app, CostructuredArrow.IsUniversal.existsUnique, AlgebraicGeometry.ι_sigmaSpec, piEquivalenceFunctorDiscrete_inverse_obj, Functor.leftKanExtensionUnit_leftKanExtension_map_leftKanExtensionObjIsoColimit_hom, Comma.fromProd_obj_hom, Limits.prod.rightUnitor_hom_naturality, Limits.Types.pi_lift_π_apply', CostructuredArrow.w_prod_fst, AlgebraicGeometry.coprodSpec_inr, CostructuredArrow.mapNatIso_inverse_obj_left, Over.epi_iff_epi_left, AlgebraicGeometry.instIsIsoSchemeSigmaSpecOfFinite, Limits.SequentialProduct.cone_π_app_comp_Pi_π_pos_assoc, Over.OverMorphism.ext_iff, StructuredArrow.map₂_obj_right, CostructuredArrow.mapIso_functor_obj_hom, Discrete.natIsoFunctor_hom_app, Limits.mapPairIso_inv_app, CommRingCat.mkUnder_ext_iff, StructuredArrow.commaMapEquivalenceCounitIso_hom_app_left_right, isCoseparator_of_isLimit_fan, SSet.hoFunctor.preservesTerminal', Profinite.Extend.cone_π_app, Classifier.SubobjectRepresentableBy.iso_inv_hom_left_comp, Discrete.monoidal_tensorUnit_as, IsGrothendieckAbelian.IsPresentable.injectivity₀.g_app, Functor.EssImageSubcategory.associator_hom_def, AlgebraicGeometry.Scheme.Cover.pullbackCoverOver'_X, Limits.prod.map_swap_assoc, instAB4StarModuleCat, MorphismProperty.Over.map_obj_left, Functor.IsLocalization.instDiscreteObjWhiskeringRightFunctorCategoryOfFiniteOfContainsIdentities, HomotopicalAlgebra.LeftHomotopyRel.exists_good_cylinder, SimplicialObject.Splitting.cofan_inj_epi_naturality_assoc, CostructuredArrow.ofCostructuredArrowProjEquivalence.inverse_obj_left_hom, Limits.instReflectsLimitsOfShapeDiscreteOfReflectsFiniteProductsOfFinite, Functor.leftExtensionEquivalenceOfIso₁_functor_obj_left, WithTerminal.coneEquiv_inverse_obj_π_app_left, Limits.asEmptyCocone_pt, Limits.prod.rightUnitor_hom_naturality_assoc, coprod_inl_leftDistrib_hom, Limits.BinaryCofan.mk_pt, Limits.MonoCoprod.mono_of_injective_aux, instAB4AddCommGrpCat, CostructuredArrow.toOver_obj_right, MorphismProperty.Over.pullbackComp_hom_app_left, Bicategory.LeftLift.w_assoc, Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.isIso_hom_app, TwoSquare.EquivalenceJ.inverse_map, Limits.MonoCoprod.binaryCofan_inl, AlgebraicGeometry.Scheme.locallyCoverDense_of_le, Limits.pushoutCoconeEquivBinaryCofan_inverse_obj, Limits.pushoutCoconeEquivBinaryCofan_functor_obj, StructuredArrow.commaMapEquivalenceUnitIso_inv_app_right_right, CostructuredArrow.ofDiagEquivalence.inverse_obj_left, Functor.RightExtension.coneAt_pt, Limits.CompleteLattice.finite_coproduct_eq_finset_sup, CostructuredArrow.eq_mk, extensiveTopology.mem_sieves_iff_contains_colimit_cofan, Limits.coprod.braiding_inv, Limits.BinaryFan.braiding_hom_fst_assoc, WithInitial.coconeEquiv_inverse_obj_pt_right, MorphismProperty.Under.mk_hom, CompHausLike.HasExplicitPullbacksOfInclusions.hasProp, Limits.Sigma.ι_isoColimit_inv, Functor.LeftExtension.postcomp₁_obj_left, Limits.pushoutCoconeEquivBinaryCofan_unitIso, Limits.FormalCoproduct.ι_comp_coproductIsoCofanPt_assoc, StructuredArrow.map₂_map_right, Discrete.functor_obj, Mono.of_coproductDisjoint, Limits.Cone.fromCostructuredArrow_map_hom, TopCat.range_prod_map, AlgebraicGeometry.instIsOpenImmersionLeftSchemeDiscretePUnitMapWalkingSpanOverTopMorphismPropertySpan, Limits.desc_op_comp_opCoproductIsoProduct'_hom, SimplicialObject.Splitting.ι_desc_assoc, Limits.preservesBinaryCoproducts_of_preservesBinaryBiproducts, AlgebraicGeometry.Scheme.Hom.toNormalization_inr_normalizationCoprodIso_hom, hasCardinalLT_arrow_discrete_iff, Limits.prod.associator_naturality_assoc, TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_right_as, toOver_obj_left, whiskerRight_coprod_inr_rightDistrib_inv_assoc, CommRingCat.toAlgHom_comp, IsGrothendieckAbelian.generatingMonomorphisms.exists_ordinal, StructuredArrow.preEquivalenceInverse_obj_right_hom, prod.rightUnitor_map, Limits.ι_colimitPointwiseProductToProductColimit_π, preserves_fin_of_preserves_binary_and_initial, ChosenPullbacksAlong.iso_mapPullbackAdj_unit_app, MorphismProperty.colimitsOfShape_le_coproducts, toOverPullbackIsoToOver_inv_app_left, Functor.ι_leftKanExtensionObjIsoColimit_inv, CostructuredArrow.w_prod_fst_assoc, Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_pt, FunctorToTypes.binaryProductEquiv_apply, Functor.LeftExtension.precomp_map_right, Over.toUnit_left, FinitaryExtensive.mono_ι, instIsClosedUnderLimitsOfShapeFunctorOppositeTypeIsIndObjectDiscreteOfHasLimitsOfShape, Functor.LeftExtension.precomp₂_obj_left, instMonoι, StructuredArrow.ofDiagEquivalence.functor_obj_right_right, FinitaryExtensive.isPullback_initial_to_binaryCofan, Functor.RightExtension.coneAt_π_app, AlgebraicGeometry.sigmaMk_mk, Limits.coprod.pentagon, ChosenPullbacksAlong.pullbackIsoOverPullback_hom_app_comp_snd_assoc, monoidalOfHasFiniteProducts.δ_eq, CostructuredArrow.mapNatIso_inverse_map_right, CostructuredArrow.unop_left_comp_ofMkLEMk_unop, Limits.Cocone.underPost_ι_app, leftAdjointOfStructuredArrowInitialsAux_apply, CommMon.EquivLaxBraidedFunctorPUnit.unitIso_hom_app_hom_hom_app, Over.braiding_inv_left, AlgebraicGeometry.Flat.instDescScheme, Localization.HasProductsOfShapeAux.adj_counit_app, Functor.RightExtension.postcomp₁_obj_left_map, MorphismProperty.Over.mapCongr_inv_app_left, CostructuredArrow.post_obj, Grothendieck.grothendieckTypeToCat_unitIso_hom_app_fiber, Functor.RightExtension.postcomp₁_map_right, FunctorToTypes.binaryProductCone_pt_obj, Over.prodLeftIsoPullback_inv_snd, whiskerLeft_coprod_inl_leftDistrib_inv, StructuredArrow.mapNatIso_functor_obj_right, Limits.piObjIso_hom_comp_π, leftAdjointOfStructuredArrowInitialsAux_symm_apply, Limits.Cofan.isColimit_iff_isIso_sigmaDesc, whiskerRight_coprod_inl_rightDistrib_inv, Discrete.sumEquiv_inverse_map, Over.iteratedSliceForwardIsoPost_inv_app, toOverUnit_map_left, CostructuredArrow.mapNatIso_unitIso_inv_app_left, instIsContinuousOverLeftDiscretePUnitIteratedSliceForwardOver, StructuredArrow.eta_hom_right, CostructuredArrow.prodFunctor_map, TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_right, Limits.CoproductsFromFiniteFiltered.liftToFinsetColimIso_aux_assoc, CostructuredArrow.map₂_obj_right, StructuredArrow.prodInverse_map, MorphismProperty.under_iff, symmetricOfHasFiniteProducts_braiding_hom, StructuredArrow.eta_inv_right, Discrete.compNatIsoDiscrete_hom_app, CartesianMonoidalCategory.fullSubcategory_isTerminalTensorUnit_lift_hom, preservesFiniteCoproductsOfPreservesBinaryAndInitial, Over.leftUnitor_hom_left, CostructuredArrow.mapIso_unitIso_hom_app_left, Limits.Types.binaryProductIso_hom_comp_snd_apply, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc, MorphismProperty.Over.pullbackMapHomPullback_app, Over.tensorObj_ext_iff, CostructuredArrow.toOver_map_right, StructuredArrow.ofDiagEquivalence.inverse_obj_right, CostructuredArrow.map_obj_hom, LightProfinite.Extend.cocone_ι_app, Comma.toPUnitIdEquiv_counitIso_hom_app, coev_expComparison, Under.mk_hom, Functor.LeftExtension.precomp_obj_hom_app, Limits.Types.binaryCoproductCocone_ι_app, BinaryCofan.isVanKampen_iff, Limits.MultispanIndex.multicoforkEquivSigmaCofork_counitIso_hom_app_hom, Over.iteratedSliceForward_forget, FinitaryExtensive.van_kampen', CategoryOfElements.to_comma_map_right, StructuredArrow.eq_mk, Comma.fromProd_obj_right, Limits.Multicofork.ofSigmaCofork_ι_app_right', Limits.BinaryFan.IsLimit.lift'_coe, toOverIsoToOverUnit_inv_app_left, MorphismProperty.Over.mk_hom, Over.ConstructProducts.conesEquivCounitIso_inv_app_hom_left, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_obj_obj_right_as, MorphismProperty.Over.pullback_obj_left, MorphismProperty.Over.map_map_left, Limits.Types.binaryProductIso_inv_comp_fst, Limits.HasBinaryBiproduct.hasLimit_pair, Over.postAdjunctionRight_counit_app, Over.conePost_obj_π_app, Discrete.id_def, leftAdjoint_preservesTerminal_of_reflective, Limits.Types.binaryProductCone_pt, GradedObject.CofanMapObjFun.inj_iso_hom, ModuleCat.HasLimit.productLimitCone_cone_π, MonoOver.mkArrowIso_hom_hom_left, Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_obj, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand_assoc, FunctorCategory.prod_preservesColimits, Functor.star_obj_as, NatTrans.instIsClosedUnderLimitsOfShapeOverFunctorEquifiberedHomDiscretePUnitOfHasCoproductsOfShapeHom, CostructuredArrow.toOver_obj_hom, Limits.Pi.map_isIso, StructuredArrow.ofStructuredArrowProjEquivalence.functor_obj_left_as, Sieve.overEquiv_le_overEquiv_iff, HasPullbacksOfInclusions.hasPullbackInr, MonoOver.map_obj_left, ChosenPullbacksAlong.snd'_left, PreGaloisCategory.card_fiber_coprod_eq_sum, ChosenPullbacksAlong.Over.tensorObj_ext_iff, Discrete.monoidalFunctorComp_isMonoidal, TopCat.sigmaCofan_ι_app, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap, Limits.Types.binaryProductLimitCone_cone, CostructuredArrow.ofDiagEquivalence.functor_obj_hom, HomotopicalAlgebra.Cylinder.LeftHomotopy.exists_good_cylinder, Over.rightUnitor_inv_left_fst, Limits.Types.binaryProductLimitCone_isLimit, Over.ConstructProducts.conesEquiv_unitIso, Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_map_hom_app, CostructuredArrow.homMk'_mk_id, Limits.Cocone.fromCostructuredArrow_ι_app, Grothendieck.grothendieckTypeToCat_unitIso_inv_app_fiber, Functor.LeftExtension.coconeAtWhiskerRightIso_inv_hom, Limits.Cone.fromStructuredArrow_π_app, WithTerminal.commaFromOver_map_left, Functor.RightExtension.precomp_map_left, ChosenPullbacksAlong.Over.tensorHom_left_snd_assoc, Bicategory.LeftExtension.ofCompId_left_as, Functor.LeftExtension.IsPointwiseLeftKanExtension.isLeftKanExtension, preservesFiniteLimits_liftToFinset, Limits.Cone.fromCostructuredArrow_obj_π, StructuredArrow.id_right, FunctorToTypes.binaryCoproductEquiv_apply, ChosenPullbacksAlong.cartesianMonoidalCategorySnd_mapPullbackAdj_unit_app, MonoOver.isIso_hom_left_iff_subobjectMk_eq, Under.pushout_map, Over.mapCongr_inv_app_left, HomotopicalAlgebra.LeftHomotopyRel.exists_very_good_cylinder, HomotopicalAlgebra.Cylinder.instIsGoodSymmOfRespectsIsoCofibrations, Limits.Types.binaryCoproductIso_inl_comp_inv, Limits.Types.binaryCoproductColimit_desc, CostructuredArrow.ofCostructuredArrowProjEquivalence.functor_obj_right_as, StructuredArrow.preEquivalenceFunctor_obj_left_as, Limits.mkCofanColimit_desc, MorphismProperty.Over.Hom.ext_iff, FreeMonoidalCategory.normalize_naturality, Limits.Types.binaryProductIso_hom_comp_fst_apply, StructuredArrow.map₂_obj_left, GrpCat.binaryProductLimitCone_cone_pt, Limits.opCoproductIsoProduct'_comp_self, Sieve.ofArrows_category, Limits.hasInitialChangeUniverse, CommMon.equivLaxBraidedFunctorPUnit_unitIso, Limits.MulticospanIndex.toPiForkFunctor_map_hom, Over.mapCongr_hom_app_left, Limits.opCoproductIsoProduct'_hom_comp_proj, Abelian.Pseudoelement.pseudoZero_iff, ChosenPullbacksAlong.Over.rightUnitor_inv_left_snd, FunctorToTypes.prodMk_snd, Discrete.equivalence_unitIso, equivToOverUnit_unitIso, CostructuredArrow.homMk'_right, Over.postCongr_inv_app_left, MorphismProperty.colimitsOfShape_discrete_le_llp_rlp, WithInitial.coconeEquiv_inverse_obj_pt_hom, Functor.ι_leftKanExtensionObjIsoColimit_inv_assoc, Over.mk_hom, Limits.pushoutCoconeEquivBinaryCofan_inverse_map_hom, ChosenPullbacksAlong.isoInv_pullback_obj_left, StructuredArrow.preEquivalence_unitIso, CostructuredArrow.ιCompGrothendieckProj_inv_app, CostructuredArrow.ofCostructuredArrowProjEquivalence.inverse_obj_hom, Limits.ι_comp_sigmaObjIso_inv, OverPresheafAux.counitForward_naturality₁, Over.mapComp_hom_app_left, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_obj_obj_hom, CostructuredArrow.mapIso_inverse_obj_right, Comma.equivProd_unitIso_hom_app_left, FunctorToTypes.binaryCoproductEquiv_symm_apply, Limits.preservesCoproductsOfShape_of_preservesBiproductsOfShape, SimplicialObject.Truncated.rightExtensionInclusion_hom_app, coprod_inl_rightDistrib_hom, MorphismProperty.Over.pullbackCongr_hom_app_left_fst, Limits.productUniqueIso_hom, monoidalOfHasFiniteCoproducts.tensorObj, Limits.BinaryBicone.toCocone_ι_app_right, Limits.Types.binaryProductFunctor_obj_map, Over.whiskerRight_left_fst, HomotopicalAlgebra.AttachCells.ofArrowIso_g₂, CostructuredArrow.w_prod_snd_assoc, Over.postAdjunctionLeft_unit_app_left, Discrete.monoidal_leftUnitor, Limits.Bicone.toCone_π_app_mk, Limits.hasColimitsOfShape_discrete, HomotopicalAlgebra.PrepathObject.symm_p, CommRingCat.pushout_inr_tensorProdObjIsoPushoutObj_inv_right_assoc, MorphismProperty.Over.mapPullbackAdj_counit_app, FGModuleCat.instFiniteCarrierSigmaObjModuleCatOfFinite, CostructuredArrow.costructuredArrowToOverEquivalence.functor_map, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_ι_isOpenImmersion, AlgebraicGeometry.Scheme.Cover.pullbackCoverOverProp_f, Functor.LeftExtension.postcompose₂_obj_right_map, Limits.PreservesColimitPair.of_iso_coprod_comparison, MonoOver.pullback_obj_arrow, Over.preservesTerminalIso_pullback, coprod_inr_rightDistrib_hom, preservesBinaryProducts_of_exponentialIdeal, TopCat.prod_topology, Over.prodLeftIsoPullback_inv_fst, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionCocone_pt, Limits.Bicone.toCocone_pt, StructuredArrow.map₂_map_left, Limits.BinaryBicone.toCone_π_app_left, TwoSquare.costructuredArrowRightwards_map, Limits.FormalCoproduct.fromIncl_comp_coproductIsoSelf_inv_assoc, Presheaf.isLimit_iff_isSheafFor_presieve, Limits.FormalCoproduct.fromIncl_comp_coproductIsoSelf_inv, WithTerminal.coneEquiv_functor_obj_π_app_star, Under.postAdjunctionRight_unit_app_right, CostructuredArrow.ofDiagEquivalence.functor_obj_left_hom, Bicategory.LeftExtension.whiskering_map, AlgebraicGeometry.coprodMk_inl, Limits.FormalCoproduct.cofan_inj, Under.map_map_right, Over.opEquivOpUnder_inverse_obj, Limits.instHasColimitsOfSizeDiscretePUnit, CostructuredArrow.mkPrecomp_left, ModuleCat.binaryProductLimitCone_cone_π_app_right, Limits.Types.coproductIso_mk_comp_inv, CostructuredArrow.mapIso_functor_map_left, StructuredArrow.prodFunctor_map, MorphismProperty.instFaithfulOverTopOverForget, TopCat.sigmaIsoSigma_inv_apply, AlgebraicTopology.DoldKan.PInfty_on_Γ₀_splitting_summand_eq_self_assoc, piEquivalenceFunctorDiscrete_functor_obj, Over.prodLeftIsoPullback_inv_snd_assoc, Functor.RightExtension.mk_hom, Over.rightUnitor_inv_left_snd, ChosenPullbacksAlong.Over.tensorObj_hom, Functor.leftExtensionEquivalenceOfIso₁_inverse_map_left, Comma.equivProd_counitIso_hom_app, AlgebraicGeometry.instIsClosedImmersionLeftSchemeDiscretePUnitOneOverSpecOf, Over.post_map, Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_isColimit_desc, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap_id, regularTopology.equalizerConditionMap_iff_nonempty_isLimit, Limits.CoproductsFromFiniteFiltered.liftToFinsetObj_obj, Pseudofunctor.presheafHom_obj, AlgebraicGeometry.instIsAffineHomDescScheme, Over.mapPullbackAdj_counit_app, CostructuredArrow.pre_obj_left, Over.iteratedSliceBackward_forget, Limits.FormalCoproduct.ι_comp_coproductIsoSelf_hom, Abelian.Pseudoelement.ModuleCat.eq_range_of_pseudoequal, Functor.leftExtensionEquivalenceOfIso₁_inverse_obj_left, Limits.Cofan.IsColimit.inj_desc, Over.postCongr_hom_app_left, CompHausLike.sigmaComparison_eq_comp_isos, Limits.Multicofork.toSigmaCofork_pt, MorphismProperty.Over.map_comp, Limits.asEmptyCone_pt, ObjectProperty.isStrongGenerator_iff_exists_extremalEpi, CartesianMonoidalCategory.ofChosenFiniteProducts.associator_naturality, Limits.braid_natural_assoc, TopCat.prodIsoProd_hom_snd_assoc, StructuredArrow.ofDiagEquivalence.inverse_obj_hom, toOverUnit_obj_left, AlgebraicGeometry.instPreservesColimitsOfShapeSchemeTopCatDiscreteForgetToTopOfSmall, Limits.pullbackConeEquivBinaryFan_functor_map_hom, CostructuredArrow.post_map, StructuredArrow.mapNatIso_functor_obj_hom, Limits.Types.Small.productIso_inv_comp_π, CostructuredArrow.mapIso_unitIso_inv_app_left, Limits.FintypeCat.productEquiv_apply, Discrete.instSubsingletonDiscreteHom, Under.map_obj_right, Limits.CofanTypes.sigma_ι_snd, ChosenPullbacksAlong.Over.tensorObj_left, StructuredArrow.mapIso_inverse_obj_right, NonPreadditiveAbelian.σ_comp, Under.post_obj, Limits.BinaryFan.braiding_hom_snd, StructuredArrow.ofDiagEquivalence.functor_obj_right_hom, AlgebraicGeometry.isPullback_inl_inl_coprodMap, StructuredArrow.mapNatIso_inverse_map_right, CostructuredArrow.map_obj_right, Over.iteratedSliceBackward_forget_forget, CostructuredArrow.proj_obj, IsSifted.colim_preservesLimits_pair_of_sSifted, Limits.Multifork.toPiFork_pt, ChosenPullbacksAlong.Over.associator_hom_left_fst, MorphismProperty.Over.instPreservesFiniteLimitsTopPullback, Over.prodLeftIsoPullback_hom_snd, CostructuredArrow.preEquivalence.functor_obj_hom, CostructuredArrow.mkPrecomp_right, HomotopicalAlgebra.AttachCells.cell_def, Limits.coprod.leftUnitor_hom, Limits.pair_obj_right, Limits.Cofan.ext_inv_hom, coprodComparison_tensorRight_braiding_hom, Functor.EssImageSubcategory.tensor_obj, Discrete.functor_map_id, FinitaryExtensive.mono_inl_of_isColimit, StructuredArrow.IsUniversal.existsUnique, Limits.pullbackConeEquivBinaryFan_counitIso, ChosenPullbacksAlong.isoInv_pullback_map_left, StructuredArrow.mapNatIso_functor_map_right, Limits.MulticospanIndex.multiforkEquivPiForkOfIsLimit_functor, SmallObject.FunctorObjIndex.comm, StructuredArrow.ofCommaSndEquivalenceInverse_map_right_left, CostructuredArrow.mapIso_functor_obj_left, AlgebraicGeometry.instIsAffineCoprodScheme, Limits.MultispanIndex.multicoforkEquivSigmaCofork_functor_map_hom, Limits.prod.leftUnitor_inv_naturality_assoc, Discrete.addMonoidal_tensorObj_as, Dial.comp_le_lemma, Over.whiskerRight_left_snd_assoc, SimplicialObject.Truncated.rightExtensionInclusion_left, Limits.FormalCoproduct.isColimitCofan_desc_f, Under.eqToHom_right, ChosenPullbacksAlong.Over.snd_eq_snd', AddCommGrpCat.HasLimit.productLimitCone_isLimit_lift, Limits.IsIPC.isIso, Limits.IsLimit.pullbackConeEquivBinaryFanFunctor_lift_left, CostructuredArrow.map_obj_left, MorphismProperty.Over.mapPullbackAdj_unit_app, CostructuredArrow.eta_hom_left, Over.associator_inv_left_fst_fst_assoc, Sieve.overEquiv_symm_iff, Limits.preservesBinaryCoproduct_of_preservesBinaryBiproduct, Sieve.functorPushforward_over_map, Limits.FormalCoproduct.ι_comp_coproductIsoSelf_hom_assoc, PreZeroHypercover.inj_sigmaOfIsColimit_f_assoc, AlgebraicGeometry.Surjective.sigmaDesc_of_union_range_eq_univ, CostructuredArrow.ofCommaFstEquivalenceFunctor_obj_hom, Over.associator_hom_left_fst, Functor.final_fromPUnit_of_isTerminal, StructuredArrow.mapNatIso_inverse_obj_left, preservesShape_fin_of_preserves_binary_and_initial, Limits.BinaryBicone.ofLimitCone_inr, StructuredArrow.ofDiagEquivalence.functor_map_right_right, CategoryOfElements.toStructuredArrow_obj, StructuredArrow.ofStructuredArrowProjEquivalence.inverse_obj_left_as, Subfunctor.equivalenceMonoOver_inverse_map, NatTrans.equifibered_of_discrete, StructuredArrow.preEquivalence_inverse, Limits.prod_rightUnitor_inv_naturality_assoc, Condensed.instPreservesLimitsOfShapeOppositeProfiniteDiscreteCarrierToTopTotallyDisconnectedSpaceOfFinite, Limits.PreservesTerminal.of_iso_comparison, FinitaryPreExtensive.isIso_sigmaDesc_map, Equalizer.Presieve.Arrows.compatible_iff, HomotopicalAlgebra.AttachCells.hm_assoc, AddCommGrpCat.binaryProductLimitCone_isLimit_lift, Limits.CoproductsFromFiniteFiltered.liftToFinset_obj_map, Comma.toPUnitIdEquiv_functor_obj, StructuredArrow.preEquivalence_functor, IsInitial.isVanKampenColimit, Limits.FormalCoproduct.coproductIsoSelf_hom_f, ChosenPullbacksAlong.Over.leftUnitor_inv_left_snd, CostructuredArrow.toStructuredArrow'_obj, Abelian.Pseudoelement.pseudoApply_mk', WithInitial.commaFromUnder_map_left, StructuredArrow.homMk'_mk_comp, NatTrans.instIsClosedUnderColimitsOfShapeUnderFunctorCoequifiberedHomDiscretePUnitOfHasProductsOfShapeHom, CostructuredArrow.prodInverse_obj, Functor.LeftExtension.postcompose₂ObjMkIso_inv_right_app, ChosenPullbacksAlong.Over.tensorUnit_hom, HomotopicalAlgebra.PathObject.exists_very_good, Over.prodLeftIsoPullback_hom_fst_assoc, CommMon.equivLaxBraidedFunctorPUnit_functor, Over.map_map_left, ChosenPullbacksAlong.Over.whiskerLeft_left_snd_assoc, MorphismProperty.Over.mapId_inv_app_left, Limits.FintypeCat.productEquiv_symm_comp_π_apply, Limits.BinaryFan.braiding_inv_snd, PreGaloisCategory.monoInducesIsoOnDirectSummand, Under.costar_obj_left, CostructuredArrow.ofDiagEquivalence.functor_obj_left_left, ChosenPullbacksAlong.Over.associator_hom_left_snd_fst, AddCommGrpCat.binaryProductLimitCone_cone_π_app_left, Comma.fromProd_obj_left, MonoOver.image_map, TopCat.prodIsoProd_inv_snd_assoc, Under.under_left, Under.mapPushoutAdj_unit_app, StructuredArrow.ofCommaSndEquivalenceFunctor_map_left, Limits.SequentialProduct.functorMap_commSq_aux, GrothendieckTopology.mem_over_iff, OverPresheafAux.counitForward_val_snd, CountableAB4.ofShape, HomotopicalAlgebra.Precylinder.symm_i_assoc, WithInitial.coconeEquiv_functor_obj_ι_app_star, Limits.FormalCoproduct.cofan_inj_φ, Over.tensorUnit_hom, TopCat.prodIsoProd_inv_fst_assoc, Over.opEquivOpUnder_inverse_map, instAB4StarAddCommGrpCat, CostructuredArrow.ofCostructuredArrowProjEquivalence.inverse_map_left_left, StructuredArrow.mkPostcomp_left, Limits.SequentialProduct.cone_π_app_comp_Pi_π_neg_assoc, StructuredArrow.left_eq_id, HomotopicalAlgebra.PathObject.ofFactorizationData_p, Over.leftUnitor_inv_left_fst, IsGrothendieckAbelian.GabrielPopescuAux.exists_d_comp_eq_d, Under.post_map, simply_connected_def, Limits.coproductUniqueIso_hom, CommAlgCat.binaryCofan_pt, preservesBinaryProducts_of_preservesTerminal_and_pullbacks, ChosenPullbacksAlong.isoInv_mapPullbackAdj_unit_app_left, Over.inv_left_hom_left, Limits.inl_of_isLimit, Over.starPullbackIsoStar_hom_app_left, CommRingCat.mkUnder_hom, StructuredArrow.mapNatIso_unitIso_hom_app_right, AlgebraicGeometry.opensDiagram_map, Limits.instHasColimitDiscreteOppositeCompInverseOppositeOpFunctor, FunctorToTypes.inr_comp_binaryCoproductIso_hom_apply, FunctorToTypes.inl_comp_binaryCoproductIso_hom_apply, CostructuredArrow.id_left, ChosenPullbacksAlong.Over.whiskerRight_left, Limits.ProductsFromFiniteCofiltered.finiteSubproductsCocone_π_app_eq_sum, Limits.ι_comp_sigmaObjIso_hom, HomotopicalAlgebra.cofibrations_over_iff, Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_ι_app, HomotopicalAlgebra.PathObject.symm_p_assoc, instAB4ModuleCat, CostructuredArrow.ofDiagEquivalence.inverse_obj_hom, WithTerminal.coneEquiv_functor_obj_π_app_of, Limits.Multicofork.ofSigmaCofork_π, Limits.MulticospanIndex.multiforkEquivPiFork_inverse_map_hom, WithInitial.coconeEquiv_unitIso_hom_app_hom_right, PreGaloisCategory.fiberBinaryProductEquiv_symm_fst_apply, SSet.instFiniteCoprod, Over.forgetMapTerminal_inv_app, Limits.FormalCoproduct.cofan_inj_f_fst, Limits.opCoproductIsoProduct'_inv_comp_inj, Limits.CofanTypes.sigma_pt, MorphismProperty.Over.mapCongr_hom_app_left, Limits.limitBiconeOfUnique_isBilimit_isColimit, Limits.Pi.isoLimit_inv_π, FunctorToTypes.binaryCoproductCocone_pt_map, prod.rightUnitorEquivalence_inverse, Subobject.inf_eq_map_pullback', Limits.MulticospanIndex.fstPiMapOfIsLimit_proj, GrpCat.binaryProductLimitCone_isLimit_lift, StructuredArrow.eqToHom_right, Limits.Cofan.inj_injective_of_isColimit, Discrete.functorComp_hom_app, Limits.Multicofork.ofSigmaCofork_pt, Presheaf.isSeparated_iff_subsingleton, TopCat.sigmaIsoSigma_hom_ι_assoc, HomotopicalAlgebra.RightHomotopyRel.exists_very_good_pathObject, Over.eqToHom_left, AlgebraicGeometry.IsZariskiLocalAtSource.sigmaDesc, Sieve.overEquiv_top, AlgebraicGeometry.isIso_stalkMap_coprodSpec, AddCommGrpCat.HasLimit.productLimitCone_cone_pt_coe, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteSpecOfFinite, Functor.essImage_underPost, Comma.equivProd_inverse_map_left, Limits.productUniqueIso_inv, Over.tensorHom_left_snd_assoc, HomotopicalAlgebra.PathObject.symm_p, Bicategory.Lan.CommuteWith.lanCompIsoWhisker_hom_right, Limits.prod.leftUnitor_hom_naturality_assoc, CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided_map_hom_hom_app, CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided_obj, FunctorToTypes.binaryProductIso_inv_comp_snd_apply, Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_hom, FunctorToTypes.binaryProductEquiv_symm_apply, prod.rightUnitorEquivalence_functor, CostructuredArrow.ιCompGrothendieckPrecompFunctorToCommaCompFst_inv_app, CartesianMonoidalCategory.preservesLimit_pair_of_isIso_prodComparison, ChosenPullbacksAlong.Over.rightUnitor_inv_left_snd_assoc, Discrete.productEquiv_counitIso_hom_app, SimplicialObject.Splitting.cofan_inj_comp_PInfty_eq_zero, TopCat.piFan_pt, Functor.LeftExtension.postcompose₂_map_right_app, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty, SimplicialObject.Splitting.ιSummand_comp_d_comp_πSummand_eq_zero, AB4StarOfSize.ofShape, Under.map_obj_hom, Limits.FormalCoproduct.inj_comp_cofanPtIsoSelf_hom_assoc, IsMonoidalLeftDistrib.preservesBinaryCoproducts_tensorLeft, CommRingCat.Under.tensorProdEqualizer_ι, Limits.Types.binaryCoproductIso_inr_comp_hom, CostructuredArrow.map_map_right, Over.leftUnitor_inv_left_snd, CostructuredArrow.closedUnderLimitsOfShape_discrete_empty, Limits.FormalCoproduct.cofan_inj_f_snd, FinitaryExtensive.mono_inr_of_isColimit, CartesianMonoidalCategory.preservesLimitsOfShape_discrete_walkingPair_of_isIso_prodComparison, ChosenPullbacksAlong.Over.leftUnitor_inv_left_fst_assoc, Limits.snd_of_isColimit, AlgebraicGeometry.Scheme.kerAdjunction_counit_app, Limits.IndObjectPresentation.toCostructuredArrow_obj_right_as, AlgebraicGeometry.opensDiagramι_app, StructuredArrow.homMk'_left, HomotopicalAlgebra.instCofibrationLeftDiscretePUnitOfOver, Over.prodComparisonIso_pullback_inv_left_snd', Limits.piObjIso_inv_comp_π, OverClass.fromOver_over, instFinalDiscreteOfIsConnected, Functor.instPreservesLimitsOfShapeDiscreteWalkingPairOfIsHomological, Functor.essImage.of_underPost, Comma.equivProd_unitIso_hom_app_right, HomologicalComplex.isSeparator_coproduct_separatingFamily, FinitaryExtensive.isVanKampen_finiteCoproducts_Fin, AlgebraicGeometry.isCompl_range_inl_inr, Limits.MulticospanIndex.multiforkEquivPiForkOfIsLimit_counitIso, PreOneHypercover.p₁_sigmaOfIsColimit, CostructuredArrow.ofCommaFstEquivalenceInverse_obj_right_as, Discrete.sumEquiv_unitIso_hom_app, MonoOver.image_obj, TopologicalSpace.Opens.coe_overEquivalence_functor_obj, IsMonoidalRightDistrib.preservesBinaryCoproducts_tensorRight, OverClass.asOver_left, AddGrpCat.binaryProductLimitCone_isLimit_lift, Over.μ_pullback_left_fst_snd', CostructuredArrow.ofCommaFstEquivalenceInverse_obj_left_left, Over.comp_left, ChosenPullbacksAlong.Over.tensorUnit_left, Over.mapPullbackAdj_unit_app, prod.leftInverseUnitor_map, MorphismProperty.Over.forget_comp_forget_map, FundamentalGroupoidFunctor.instIsIsoFanGrpdObjTopCatFundamentalGroupoidFunctorPiTopToPiCone, TopCat.piIsoPi_inv_π_apply, LightCondensed.instPreservesEpimorphismsFunctorDiscreteNatLightCondModLim, Functor.RightExtension.postcomp₁_obj_left_obj, CostructuredArrow.mapNatIso_functor_obj_hom, Limits.MulticospanIndex.multiforkEquivPiFork_functor_obj_pt, Over.mapId_inv_app_left, CostructuredArrow.w_prod_snd, MorphismProperty.Over.pullbackComp_inv_app_left, CommRingCat.toAlgHom_id, Comma.toIdPUnitEquiv_unitIso_inv_app_right, Limits.PreservesCoproduct.of_iso_comparison, StructuredArrow.mono_iff_mono_right, CostructuredArrow.grothendieckPrecompFunctorToComma_obj_left, Ind.isSeparator_range_yoneda, Comma.toPUnitIdEquiv_unitIso_inv_app_left, Limits.Types.binaryCoproductIso_inl_comp_hom_apply, ChosenPullbacksAlong.Over.leftUnitor_inv_left_fst, Sieve.overEquiv_symm_pullback, CostructuredArrow.CreatesConnected.natTransInCostructuredArrow_app, Subfunctor.equivalenceMonoOver_inverse_obj, Over.leftUnitor_inv_left_snd_assoc, Functor.LeftExtension.postcompose₂_obj_hom_app, HasPullbacksOfInclusions.hasPullbackInl, instIsLocallyDirectedDiscrete, Under.mono_iff_mono_right, TopCat.Presheaf.SheafConditionEqualizerProducts.piInters.hom_ext_iff, WithInitial.liftFromUnder_obj_obj, TopCat.isInducing_prodMap, CategoryOfElements.costructuredArrowYonedaEquivalence_counitIso, StructuredArrow.ofStructuredArrowProjEquivalence.functor_map_right_right, HomotopicalAlgebra.Cylinder.ofFactorizationData_i, FunctorToTypes.inl_comp_binaryCoproductIso_hom, AlgebraicGeometry.instIsOpenImmersionInrScheme, CountableAB4Star.ofShape, IsPushout.of_isColimit_binaryCofan_of_isInitial, TopCat.Presheaf.whiskerIsoMapGenerateCocone_inv_hom, StructuredArrow.ofCommaSndEquivalenceFunctor_obj_right, Limits.MultispanIndex.ofSigmaCoforkFunctor_obj, Over.ConstructProducts.conesEquivFunctor_map_hom, MorphismProperty.instHasPullbackHomDiscretePUnitOfHasPullbacksAlong, ChosenPullbacksAlong.iso_pullback_obj, Limits.BinaryFan.rightUnitor_inv, Discrete.equivalence_functor, HomologicalComplex.instHasLimitDiscreteWalkingPairCompPairEval, FreeBicategory.preinclusion_map₂, Limits.MulticospanIndex.fstPiMapOfIsLimit_proj_assoc, StructuredArrow.hom_eq_iff, PreGaloisCategory.nonempty_fiber_pi_of_nonempty_of_finite, Limits.BinaryCofan.isColimit_iff_isIso_inl, Under.postEquiv_counitIso, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionCocone_ι_app, WithTerminal.coneEquiv_inverse_obj_pt_left, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand₀, CommRingCat.mkUnder_right, Over.conePost_map_hom, piEquivalenceFunctorDiscrete_unitIso, Under.postComp_hom_app_right, FunctorToTypes.binaryProductIso_hom_comp_snd, Functor.Monoidal.μ_of_cartesianMonoidalCategory, LightCondensed.instPreservesLimitsOfShapeOppositeLightProfiniteDiscreteCarrier, Limits.FormalCoproduct.cofanHomEquiv_symm_apply_φ, FreeMonoidalCategory.tensorFunc_map_app, Presieve.preservesTerminal_of_isSheaf_for_empty, Limits.PreservesFiniteProducts.preserves, Limits.Sigma.ι_isoColimit_hom_assoc, MonoOver.bot_left, SSet.hoFunctor.preservesTerminal, SSet.Truncated.rightExtensionInclusion_right_as, Presieve.ofArrows_category, WithInitial.coconeEquiv_functor_map_hom, FinitaryPreExtensive.hasPullbacks_of_is_coproduct, Functor.RightExtension.coneAtWhiskerRightIso_inv_hom, HomotopicalAlgebra.PathObject.instIsVeryGoodSymmOfRespectsIsoFibrations, FGModuleCat.instFiniteCarrierPiObjModuleCatOfFinite, AlgebraicGeometry.disjoint_opensRange_sigmaι, CostructuredArrow.mapNatIso_counitIso_inv_app_left, OverPresheafAux.counitAuxAux_inv, Functor.LeftExtension.postcomp₁_obj_hom_app, SmallObject.functorMapSrc_functorObjTop, PreGaloisCategory.FiberFunctor.preservesTerminalObjects, StructuredArrow.mk_left, TopologicalSpace.Opens.overEquivalence_unitIso_hom_app_left, MonoOver.map_obj_arrow, SmallObject.functorObj_comm, AlgebraicTopology.DoldKan.N₂Γ₂_inv_app_f_f, Limits.opProductIsoCoproduct'_comp_self, CategoryOfElements.fromStructuredArrow_obj, Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_map, StructuredArrow.ofDiagEquivalence.functor_obj_right_left_as, Limits.pullbackConeEquivBinaryFan_inverse_obj, ChosenPullbacksAlong.cartesianMonoidalCategoryFst_mapPullbackAdj_unit_app, CostructuredArrow.mapIso_counitIso_hom_app_left, CostructuredArrow.mk_right, Limits.pullbackConeEquivBinaryFan_functor_obj, hasExactColimitsOfShape_discrete_finite, OverPresheafAux.restrictedYonedaObj_map, WithInitial.coconeEquiv_inverse_obj_pt_left_as, Over.map_obj_hom, Limits.Types.Small.productIso_inv_comp_π_apply, CostructuredArrow.mapCompιCompGrothendieckProj_inv_app, AlgebraicGeometry.ι_left_coprodIsoSigma_inv, SimplicialObject.Splitting.ofIso_isColimit', Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_laxMonoidalToMon_obj_one, Over.associator_hom_left_snd_fst, Limits.Types.binaryCoproductIso_inr_comp_hom_apply, CostructuredArrow.toOver_map_left, Over.postComp_inv_app_left, Limits.FormalCoproduct.inj_comp_cofanPtIsoSelf_hom, Bicategory.LeftExtension.whiskerIdCancel_right, Limits.FormalCoproduct.fromIncl_comp_cofanPtIsoSelf_inv_assoc, Limits.prod.triangle, Limits.Cofan.ext_hom_hom, RanIsSheafOfIsCocontinuous.fac', Limits.IndObjectPresentation.yoneda_isColimit_desc, Limits.coprod.map_swap, Limits.instHasLimitOppositeDiscreteOpFunctor, NonPreadditiveAbelian.epi_r, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscreteWalkingPairSpec, Limits.Bicone.ofColimitCocone_ι, subterminalsEquivMonoOverTerminal_inverse_map, whiskerLeft_coprod_inl_leftDistrib_inv_assoc, FinitaryExtensive.vanKampen, CostructuredArrow.eta_inv_left, MorphismProperty.over_iso_iff, Limits.MonoCoprod.mono_of_injective, Limits.MulticospanIndex.multiforkEquivPiFork_counitIso_inv_app_hom, SmallObject.ιFunctorObj_eq, Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidalObj_obj, Functor.leftExtensionEquivalenceOfIso₁_inverse_map_right, Functor.equiv_functor_obj, NonPreadditiveAbelian.isIso_r, underToAlgebra_obj_A, Limits.Cofan.mk_ι_app, Limits.MulticospanIndex.parallelPairDiagramOfIsLimit_map, Functor.toOver_obj_left, SimplyConnectedSpace.equiv_unit, TwoSquare.costructuredArrowDownwardsPrecomp_obj, AlgEquiv.toUnder_inv_right_apply, Limits.BinaryCofan.isColimit_iff_isIso_inr, CategoryOfElements.fromCostructuredArrow_obj_snd, Functor.LeftExtension.postcompose₂_obj_right_obj, AlgebraicGeometry.IsZariskiLocalAtTarget.coprodMap, AlgebraicGeometry.IsFinite.instDescScheme, CostructuredArrow.mapIso_functor_map_right, StructuredArrow.ofStructuredArrowProjEquivalence.functor_obj_right_hom, prod.rightUnitorEquivalence_counitIso, HomotopicalAlgebra.PathObject.instIsGoodTrans, Limits.Multicofork.ofSigmaCofork_ι_app_right, Over.rightUnitor_inv_left_snd_assoc, Over.toOverSectionsAdj_counit_app, AlgebraicGeometry.Scheme.Hom.inl_normalizationCoprodIso_hom_fromNormalization_assoc, Functor.relativelyRepresentable.diag_iff, StructuredArrow.toCostructuredArrow_map, Limits.instHasLimitsOfSizeDiscretePUnit, ChosenPullbacksAlong.Over.whiskerRight_left_snd, Mono.cofanInr_of_binaryCoproductDisjoint, AlgebraicGeometry.nonempty_isColimit_cofanMk_of, Limits.ReflectsFiniteProducts.reflects, MonoOver.w, MonoOver.bot_arrow_eq_zero, NatTrans.Equifibered.of_discrete, GradedObject.CofanMapObjFun.ιMapObj_iso_inv, SimplicialObject.Splitting.πSummand_comp_cofan_inj_id_comp_PInfty_eq_PInfty, AlgebraicGeometry.Scheme.Cover.pullbackCoverOverProp'_X, ObjectProperty.IsClosedUnderFiniteProducts.isClosedUnderLimitsOfShape, forgetAdjToOver_unit_app, Functor.initial_fromPUnit_of_isInitial, OverPresheafAux.counitBackward_counitForward, Limits.Types.binaryProductFunctor_map_app, Over.iteratedSliceEquivOverMapIso_inv_app_left_left, SimplicialObject.Split.toKaroubiNondegComplexFunctorIsoN₁_hom_app_f_f, CostructuredArrow.mapNatIso_inverse_map_left, Over.iteratedSliceEquivOverMapIso_hom_app_left_left, CostructuredArrow.costructuredArrowToOverEquivalence.functor_obj, SimplicialObject.Splitting.cofan_inj_πSummand_eq_id, Comma.equivProd_counitIso_inv_app, CostructuredArrow.map₂_map_right, Comma.toIdPUnitEquiv_unitIso_hom_app_right, FunctorToTypes.binaryProductLimit_lift, AlgebraicGeometry.HasAffineProperty.coprodDesc_affineAnd, CommRingCat.toAlgHom_apply, StructuredArrow.toUnder_map_right, Mono.cofanInl_of_binaryCoproductDisjoint, regularTopology.parallelPair_pullback_initial, WithTerminal.liftFromOver_obj_obj, Limits.biproduct.conePointUniqueUpToIso_inv, overToCoalgebra_map_f, Limits.IndObjectPresentation.toCostructuredArrow_obj_left, Pseudofunctor.isStackFor_iff, StructuredArrow.IsUniversal.hom_desc, CostructuredArrow.ofCostructuredArrowProjEquivalence.functor_obj_left_right_as, CommRingCat.pushout_inl_tensorProdObjIsoPushoutObj_inv_right_assoc, Limits.Cocone.fromStructuredArrow_obj_pt, Pseudofunctor.isPrestackFor_iff_isSheafFor, Bicategory.LeftExtension.IsKan.uniqueUpToIso_hom_right, CartesianMonoidalCategory.ofChosenFiniteProducts.leftUnitor_naturality, Limits.CoproductsFromFiniteFiltered.liftToFinsetObj_map, ChosenPullbacksAlong.Over.whiskerRight_left_snd_assoc, FinitaryExtensive.isVanKampen_finiteCoproducts, HomotopicalAlgebra.RightHomotopyRel.exists_good_pathObject, CostructuredArrow.pre_obj_right, CommMon.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon_obj, Limits.limitConeOfUnique_cone_pt, AlgebraicGeometry.Scheme.Cover.pullbackCoverOverProp'_f, Limits.Cofan.mk_pt, Limits.piObjIso_hom_comp_π_assoc, ChosenPullbacksAlong.Over.whiskerRight_left_fst_assoc, Over.ConstructProducts.conesEquivUnitIso_inv_app_hom, TwoSquare.isIso_lanBaseChange_app_iff, Over.pullback_map_left, HomotopicalAlgebra.Cylinder.exists_very_good, instIsContinuousOverLeftDiscretePUnitIteratedSliceBackwardOver, Under.forgetMapInitial_hom_app, Presieve.isSheafFor_of_preservesProduct, CostructuredArrow.CreatesConnected.raiseCone_pt, AlgebraicTopology.DoldKan.Γ₀.Obj.mapMono_on_summand_id, FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin, Limits.Types.binaryCoproductIso_inr_comp_inv_apply, Discrete.sumEquiv_counitIso_hom_app, Limits.pushoutCoconeEquivBinaryCofan_functor_map_hom, Over.prodComparisonIso_pullback_inv_left_fst_fst, Pseudofunctor.presheafHom_map, FreeMonoidalCategory.inclusion_map, MorphismProperty.costructuredArrow_iso_iff, Discrete.instIsIso, Limits.IndObjectPresentation.toCostructuredArrow_map_left, AlgebraicGeometry.Scheme.Hom.inl_normalizationCoprodIso_hom_fromNormalization, WithTerminal.coneEquiv_inverse_obj_pt_hom, Over.postEquiv_counitIso, Limits.Fan.ext_inv_hom, WithInitial.coconeEquiv_inverse_obj_ι_app_right, Limits.Bicone.toCone_pt, Comma.toIdPUnitEquiv_functor_map, Over.ConstructProducts.conesEquivCounitIso_hom_app_hom_left, Functor.ranObjObjIsoLimit_inv_π_assoc, MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso_hom_app_app, Over.sections_obj, AlgebraicGeometry.Scheme.smallGrothendieckTopologyOfLE_eq_toGrothendieck_smallPretopology, Functor.RightExtension.precomp_obj_right, SimplicialObject.Split.cofan_inj_naturality_symm_assoc, Discrete.monoidalFunctor_ε, Limits.coprod.leftUnitor_inv, Comma.toIdPUnitEquiv_inverse_obj_left_as, AlgebraicGeometry.opensDiagram_obj, CommRingCat.coproductCocone_pt, Limits.PreservesFiniteLimitsOfIsFilteredCostructuredArrowYonedaAux.isoAux_hom_app, Limits.BinaryBicone.ofColimitCocone_fst, Mon.equivLaxMonoidalFunctorPUnit_counitIso, CostructuredArrow.toStructuredArrow'_map, MorphismProperty.instFullUnderTopUnderForget, Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_hom_assoc, Mon.equivLaxMonoidalFunctorPUnit_unitIso, Limits.Fan.ext_hom_hom, Functor.leftExtensionEquivalenceOfIso₁_inverse_obj_right, Over.star_obj_left, Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_obj, isSeparator_iff_of_isColimit_cofan, Discrete.functor_obj_eq_as, Over.iteratedSliceEquiv_functor, WithInitial.coconeEquiv_functor_obj_ι_app_of, MorphismProperty.overObj_iff, AlgebraicGeometry.instIsOpenImmersionMapScheme, Limits.BinaryFan.π_app_right, AlgebraicGeometry.sigmaι_eq_iff, IsGrothendieckAbelian.ab4OfSize, AlgebraicGeometry.opensCone_π_app, Over.tensorHom_left, CostructuredArrow.w, Dial.Hom.le, discreteEquiv_symm_apply_as, MorphismProperty.instIsLeftAdjointOverTopMapOfHasPullbacksAlongOfIsStableUnderBaseChangeAlong, CostructuredArrow.ofCostructuredArrowProjEquivalence.inverse_obj_right_as, StructuredArrow.commaMapEquivalenceInverse_obj, Limits.BinaryCofan.ι_app_right, MorphismProperty.Over.mapComp_hom_app_left, toOverIteratedSliceForwardIsoPullback_hom_app_left, Mon.EquivLaxMonoidalFunctorPUnit.unitIso_hom_app_hom_app, StructuredArrow.homMk'_id, Functor.RightExtension.IsPointwiseRightKanExtension.isIso_hom, MonoOver.mk'_coe', Functor.RightExtension.mk_right_as, Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_inv_assoc, prod.leftInverseUnitor_obj, StructuredArrow.w_prod_snd, Functor.leftExtensionEquivalenceOfIso₁_counitIso_inv_app_right_app, Limits.MultispanIndex.multicoforkEquivSigmaCofork_functor_obj_pt, Limits.MultispanIndex.inj_fstSigmaMapOfIsColimit_assoc, StructuredArrow.ofStructuredArrowProjEquivalence.inverse_obj_hom, StructuredArrow.preEquivalenceInverse_obj_right_right, CostructuredArrow.ofCostructuredArrowProjEquivalence.functor_obj_left_hom, Presheaf.isSheaf_iff_isLimit, Under.opEquivOpOver_functor_obj, CechNerveTerminalFrom.wideCospan.limitIsoPi_inv_comp_pi, Limits.instPreservesLimitsOfShapeDiscreteOfFiniteOfPreservesFiniteProducts, StructuredArrow.preEquivalenceInverse_obj_right_left_as, Over.ConstructProducts.conesEquiv_counitIso, Comma.toIdPUnitEquiv_functor_obj, MonoOver.mono, Bicategory.Lan.CommuteWith.lanCompIsoWhisker_inv_right, StructuredArrow.mapNatIso_inverse_obj_hom, Over.associator_inv_left_fst_snd, Limits.Fan.IsLimit.fac_assoc, AlgebraicGeometry.instSurjectiveDescI₀SchemeF, Discrete.addMonoidalFunctor_obj, Functor.LeftExtension.postcompose₂ObjMkIso_hom_right_app, CechNerveTerminalFrom.wideCospan.limitIsoPi_hom_comp_pi_assoc, coprod_inl_leftDistrib_hom_assoc, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.cocone_ι_transitionMap_assoc, MonoOver.forget_obj_left, WithInitial.commaFromUnder_obj_hom_app, any_functor_const_on_obj, StructuredArrow.commaMapEquivalenceFunctor_obj_left, CostructuredArrow.mapNatIso_inverse_obj_hom, Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_pt, Mon.EquivLaxMonoidalFunctorPUnit.counitIso_inv_app_hom, Limits.Types.binaryCoproductIso_inr_comp_inv, Over.forget_obj, CostructuredArrow.grothendieckProj_map, Limits.Bicone.ofLimitCone_ι, MorphismProperty.Over.w_assoc, toOverPullbackIsoToOver_hom_app_left, TopCat.prodIsoProd_hom_snd, Over.star_obj_hom, Functor.RightExtension.postcomp₁_map_left_app, PreGaloisCategory.fiberBinaryProductEquiv_symm_snd_apply, prod.leftUnitorEquivalence_inverse, Limits.instIsIPCFunctor, Comma.toIdPUnitEquiv_counitIso_hom_app, Under.forget_map, PreservesPullbacksOfInclusions.preservesPullbackInl, HomotopicalAlgebra.PathObject.instIsVeryGoodOfFactorizationData, Limits.preservesInitial_of_isIso, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand₀'_assoc, NonPreadditiveAbelian.mono_Δ, Over.associator_hom_left_snd_snd_assoc, commAlgCatEquivUnder_counitIso, AlgebraicGeometry.Scheme.Hom.inr_normalizationCoprodIso_hom_fromNormalization_assoc, SimplicialObject.Truncated.rightExtensionInclusion_right_as, StructuredArrow.post_map, StructuredArrow.commaMapEquivalenceFunctor_map_left, CartesianMonoidalCategory.ofChosenFiniteProducts.triangle, monoidalOfHasFiniteCoproducts.whiskerRight, prod.rightUnitor_isEquivalence, SmallObject.ι_functorMapSrc_assoc, Functor.LeftExtension.postcomp₁_obj_right_map, Limits.pointwiseProductCompEvaluation_inv_app, Functor.ranObjObjIsoLimit_hom_π_assoc, Limits.IndObjectPresentation.yoneda_ι_app, TopologicalSpace.Opens.overEquivalence_counitIso_inv_app, Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidalObj_map, Discrete.instSubsingleton, Comma.toPUnitIdEquiv_inverse_map_left, Under.equivalenceOfIsInitial_unitIso, hoFunctor.preservesBinaryProducts, Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidalObj_μ, Limits.BinaryFan.braiding_inv_fst_assoc, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd, coprod_inr_leftDistrib_hom, ChosenPullbacksAlong.pullbackIsoOverPullback_inv_app_comp_fst, HomotopicalAlgebra.AttachCells.reindex_cofan₂, Bicategory.LeftLift.whiskerHom_right, AlgebraicGeometry.coprodSpec_coprodMk, CostructuredArrow.epi_left_of_epi, TopCat.Presheaf.generateEquivalenceOpensLe_functor, instIsEmptyDiscrete, Limits.preservesTerminal_of_iso, Functor.instReflectsIsomorphismsDiscreteObjWhiskeringLeftIncl, Limits.BinaryBicone.toCone_pt, Bicategory.LeftExtension.ofCompId_hom, Limits.Multifork.toPiFork_π_app_one, Limits.Multifork.ofPiFork_pt, instExtensiveOfArrowsι, Limits.Types.productIso_hom_comp_eval_apply, SimplicialObject.Splitting.cofan_inj_πSummand_eq_zero, Over.μ_pullback_left_fst_fst', ChosenPullbacksAlong.cartesianMonoidalCategoryToUnit_pullback_obj, CostructuredArrow.ofCommaFstEquivalenceFunctor_map_right, Types.monoOverEquivalenceSet_functor_map, Limits.asEmptyCocone_ι_app, Over.μ_pullback_left_fst_snd, Over.whiskerRight_left, Functor.EssImageSubcategory.toUnit_def, CostructuredArrow.homMk'_left, Limits.biprod.conePointUniqueUpToIso_hom, TopologicalSpace.Opens.overEquivalence_unitIso_inv_app_left, StructuredArrow.homMk'_right, CategoryOfElements.costructuredArrowULiftYonedaEquivalence_counitIso, Limits.IndObjectPresentation.toCostructuredArrow_obj_hom, StructuredArrow.pre_map_left, Limits.Fan.IsLimit.fac, CommRingCat.pushout_inl_tensorProdObjIsoPushoutObj_inv_right, Limits.colimitPointwiseProductToProductColimit_app, CostructuredArrow.ofDiagEquivalence.functor_obj_right_as, NonPreadditiveAbelian.lift_sub_lift, AlgebraicGeometry.instHasColimitsOfShapeDiscreteSchemeOfSmall, Over.opEquivOpUnder_counitIso, TopCat.Presheaf.SheafConditionEqualizerProducts.res_π, CostructuredArrow.prodFunctor_obj, AddCommGrpCat.HasLimit.productLimitCone_cone_π, TopCat.prodIsoProd_inv_fst, Functor.LeftExtension.coconeAtWhiskerRightIso_hom_hom, Comma.equivProd_inverse_map_right, OverPresheafAux.counitAux_hom, Functor.RightExtension.IsPointwiseRightKanExtensionAt.isIso_hom_app, CostructuredArrow.mapNatIso_unitIso_hom_app_left, commAlgCatEquivUnder_inverse_obj_carrier, MorphismProperty.instFaithfulCostructuredArrowTopOverToOver, Functor.equiv_unitIso, SmallObject.ι_functorMapTgt, uncurry_expComparison, Functor.leftExtensionEquivalenceOfIso₁_inverse_obj_hom_app, CategoryOfElements.fromCostructuredArrow_map_coe, instSmallDiscrete, CommRingCat.Under.instPreservesLimitUnderDiscreteFunctorTensorProdOfFinite, Discrete.natIso_app, Limits.prod.associator_hom, Functor.preservesInitialObject_of_preservesZeroMorphisms, CostructuredArrow.proj_map, AlgebraicTopology.DoldKan.PInfty_on_Γ₀_splitting_summand_eq_self, Limits.BinaryFan.braiding_hom_fst, Bicategory.LeftLift.ofIdComp_right, Limits.Fan.mk_pt, Discrete.equivOfEquivalence_apply, extendCofan_pt, CommMon.equivLaxBraidedFunctorPUnit_inverse, StructuredArrow.map_map_left, GradedObject.CofanMapObjFun.inj_iso_hom_assoc, Functor.instPreservesLimitOfIsCoreflexivePairDiscreteObjWhiskeringLeftIncl, Comma.toPUnitIdEquiv_functor_iso, ChosenPullbacksAlong.pullbackIsoOverPullback_inv_app_comp_snd, TwoSquare.structuredArrowRightwardsOpEquivalence.functor_map_left_right, FinitaryExtensive.isPullback_initial_to_sigma_ι, Discrete.addMonoidal_rightUnitor, StructuredArrow.mkPostcomp_right, CostructuredArrow.ofCommaFstEquivalenceInverse_map_left_left, Functor.LeftExtension.precomp_obj_right, toOverIsoToOverUnit_hom_app_left, rightAdjointOfCostructuredArrowTerminalsAux_apply, SmallObject.ρFunctorObj_π, prod.rightUnitor_obj, Functor.preservesBinaryProduct_of_preservesKernels, Over.w, Limits.mkFanLimit_lift, MorphismProperty.CostructuredArrow.toOver_map, Mon.EquivLaxMonoidalFunctorPUnit.counitIsoAux_inv, StructuredArrow.mapIso_functor_obj_right, hasExactLimitsOfShape_discrete_of_hasExactLimitsOfShape_finset_discrete_op, AlgebraicGeometry.instHasCoproductsOfShapeOverSchemeTopMorphismPropertyOfSmall, FreeMonoidalCategory.normalizeIsoApp_unitor, ChosenPullbacksAlong.cartesianMonoidalCategorySnd_pullback_map, Limits.SequentialProduct.cone_π_app_comp_Pi_π_pos, TopCat.Presheaf.generateEquivalenceOpensLe_counitIso, StructuredArrow.mapIso_unitIso_inv_app_right, FunctorToTypes.binaryCoproductColimit_desc, StructuredArrow.prodFunctor_obj, HomotopicalAlgebra.PathObject.RightHomotopy.exists_good_pathObject, Limits.MulticospanIndex.multiforkEquivPiFork_functor_obj_π_app, AlgHom.toUnder_right, SimplicialObject.Splitting.cofan_inj_comp_app, TopCat.range_pullback_to_prod, StructuredArrow.proj_map, Functor.LeftExtension.postcompose₂_obj_left, Sieve.yonedaFamily_fromCocone_compatible, subterminalsEquivMonoOverTerminal_inverse_obj_obj, OverClass.asOverHom_left, CostructuredArrow.comp_left, Mon.equivLaxMonoidalFunctorPUnit_functor, ChosenPullbacksAlong.Over.whiskerLeft_left_fst_assoc, coalgebraEquivOver_counitIso, Alexandrov.lowerCone_π_app, Limits.Cofan.IsColimit.inj_desc_assoc, extendCofan_ι_app, Over.closedUnderLimitsOfShape_discrete_empty, Functor.LeftExtension.IsPointwiseLeftKanExtension.isIso_hom, Functor.RightExtension.IsPointwiseRightKanExtensionAt.isoLimit_inv_π, Functor.LeftExtension.precomp_obj_left, Functor.preservesCoproduct_of_preservesCokernels, StructuredArrow.ofStructuredArrowProjEquivalence.functor_obj_right_right, HomotopicalAlgebra.Cylinder.symm_i, OverPresheafAux.unitAuxAux_inv_app_snd_coe, hoFunctor.isIso_prodComparison_stdSimplex, MorphismProperty.CostructuredArrow.toOver_obj, Limits.instHasLimitDiscreteOppositeCompInverseOppositeOpFunctor, Types.monoOverEquivalenceSet_functor_obj, MorphismProperty.Under.w, StructuredArrow.mapIso_functor_map_left, Limits.Types.productIso_hom_comp_eval, StructuredArrow.mono_right_of_mono, ModuleCat.binaryProductLimitCone_cone_π_app_left, Equalizer.Presieve.Arrows.compatible_iff_of_small, AlgEquiv.toUnder_hom_right_apply, AlgebraicGeometry.Scheme.kerFunctor_map, hasExactColimitsOfShape_discrete_of_hasExactColimitsOfShape_finset_discrete, CostructuredArrow.mk_left, SSet.Truncated.rightExtensionInclusion_left, FundamentalGroupoid.punitEquivDiscretePUnit_unitIso, Limits.HasFiniteCoproducts.out, Pseudofunctor.IsStack.essSurj_of_sieve, HomotopicalAlgebra.PrepathObject.symm_p_assoc, CostructuredArrow.homMk'_mk_comp, Limits.limitBiconeOfUnique_isBilimit_isLimit, Over.associator_inv_left_fst_fst, AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization, Limits.Multicofork.toSigmaCofork_π, Functor.leftExtensionEquivalenceOfIso₁_functor_obj_hom_app, StructuredArrow.ofStructuredArrowProjEquivalence.inverse_map_right_right, Limits.MultispanIndex.multicoforkEquivSigmaCofork_inverse_obj_pt, CostructuredArrow.right_eq_id, Limits.Sigma.ι_isoColimit_hom, Limits.preservesCoproduct_of_preservesBiproduct, StructuredArrow.comp_right, CostructuredArrow.ofDiagEquivalence.inverse_obj_right_as, MonoOver.isIso_iff_isIso_hom_left, StructuredArrow.commaMapEquivalenceFunctor_map_right, Over.snd_left, CostructuredArrow.ofCostructuredArrowProjEquivalence.functor_map_left_left, ChosenPullbacksAlong.pullbackIsoOverPullback_hom_app_comp_snd, AlgebraicGeometry.instIsOpenImmersionInlScheme, SmallObject.functorMap_comm, MorphismProperty.coproducts_iff, CostructuredArrow.prodInverse_map, Localization.preservesProductsOfShape, ChosenPullbacksAlong.Over.tensorHom_left_fst_assoc, Discrete.addMonoidal_leftUnitor, Over.over_right, instIsCocontinuousOverLeftDiscretePUnitIteratedSliceBackwardOver, Comma.fromProd_map_right, Limits.prod.associator_inv, commAlgCatEquivUnder_unitIso, WithTerminal.widePullbackShapeEquiv_inverse_obj, Over.tensorHom_left_fst, Sieve.overEquiv_symm_generate, TopCat.piIsoPi_hom_apply, Limits.MulticospanIndex.toPiForkFunctor_obj, discreteCountable, Comma.equivProd_inverse_obj_right, MorphismProperty.Over.mk_left, Over.whiskerRight_left_snd, AlgebraicGeometry.Scheme.Cover.pullbackCoverOver_f, ChosenPullbacksAlong.cartesianMonoidalCategoryToUnit_mapPullbackAdj_unit_app, FreeMonoidalCategory.instSubsingletonHomCompDiscreteNormalMonoidalObject, GlueData.π_epi, AlgebraicGeometry.isOpenImmersion_sigmaDesc, prod.leftUnitorEquivalence_counitIso, MorphismProperty.structuredArrowObj_iff, Limits.FormalCoproduct.instPreservesLimitOppositeDiscreteFunctorCompOpObjFunctorEvalOp, TopCat.Presheaf.whiskerIsoMapGenerateCocone_hom_hom, Discrete.natIso_hom_app, AlgebraicGeometry.coprodSpec_inr_assoc, ChosenPullbacksAlong.Over.associator_inv_left_fst_snd_assoc, ChosenPullbacksAlong.Over.associator_inv_left_fst_snd, WithTerminal.commaFromOver_obj_hom_app, WithTerminal.subsingleton_hom, StructuredArrow.map_obj_left, AlgebraicGeometry.Scheme.Cover.toPresieveOver_le_arrows_iff, monoidalOfHasFiniteCoproducts.rightUnitor_hom, Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_pt, MorphismProperty.CostructuredArrow.mk_left, Limits.BinaryBicone.ofColimitCocone_pt, Limits.ProductsFromFiniteCofiltered.liftToFinset_map_app, Functor.RightExtension.IsPointwiseRightKanExtension.isRightKanExtension, Limits.MonoCoprod.mono_inl_iff, HomotopicalAlgebra.instFibrationLeftDiscretePUnitOfOver, instFiniteDiscrete, Over.iteratedSliceEquiv_unitIso, Classifier.SubobjectRepresentableBy.iso_inv_left_π_assoc, underToAlgebra_obj_a, Comma.toPUnitIdEquiv_unitIso_hom_app_left, HomotopicalAlgebra.AttachCells.reindex_cofan₁, CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_μ, Limits.FormalCoproduct.fromIncl_comp_cofanPtIsoSelf_inv, Limits.asEmptyCone_π_app, Comma.fromProd_map_left, prod.rightUnitorEquivalence_unitIso, Limits.MulticospanIndex.multiforkEquivPiForkOfIsLimit_inverse, Limits.Types.binaryProductIso_inv_comp_snd, WithInitial.commaFromUnder_map_right, IsPreconnected.iso_constant, Limits.Types.binaryProductIso_hom_comp_fst, Limits.Bicone.toCocone_ι_app, ChosenPullbacksAlong.Over.whiskerRight_left_fst, Limits.MulticospanIndex.multiforkEquivPiFork_inverse_obj_π_app, Under.mk_right, StructuredArrow.ofCommaSndEquivalenceInverse_obj_hom, SmallObject.functorMapSrc_functorObjTop_assoc, CostructuredArrow.toStructuredArrow_obj, Functor.toOver_map_left, Limits.Pi.map_eq_prod_map, Limits.ι_comp_sigmaObjIso_hom_assoc, AlgebraicGeometry.instQuasiCompactLiftSchemeIdOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, FunctorToTypes.inr_comp_binaryCoproductIso_hom, Over.star_map_left, Over.tensorHom_left_fst_assoc, ChosenPullbacksAlong.Over.associator_inv_left_snd_assoc, forgetAdjToOver.homEquiv_symm, Limits.MultispanIndex.inj_sndSigmaMapOfIsColimit_assoc, Limits.Types.binaryProductIso_hom_comp_snd, Under.postMap_app, AlgebraicGeometry.Scheme.Cover.overEquiv_generate_toPresieveOver_eq_ofArrows, ObjectProperty.instIsClosedUnderLimitsOfShapeDiscreteOfIsClosedUnderFiniteProductsOfFinite, OverPresheafAux.restrictedYonedaObj_obj, hasExactLimitsOfShape_discrete_finite, ContinuousMap.piComparison_fac, Functor.Monoidal.ε_of_cartesianMonoidalCategory, Limits.BinaryBicone.ofColimitCocone_inr, CostructuredArrow.ofCostructuredArrowProjEquivalence.functor_obj_hom, CostructuredArrow.CreatesConnected.raiseCone_π_app, AlgebraicGeometry.Scheme.isSeparated_iff_isClosedImmersion_prod_lift, extendFan_π_app, Discrete.equivalence_counitIso, CommGrpCat.binaryProductLimitCone_cone_pt, Over.prodLeftIsoPullback_inv_fst_assoc, PreOneHypercover.sigmaOfIsColimit_Y, Bicategory.LeftExtension.IsKan.uniqueUpToIso_inv_right, ChosenPullbacksAlong.Over.rightUnitor_hom_left, Discrete.sumEquiv_functor_obj, Discrete.productEquiv_counitIso_inv_app, MorphismProperty.Under.Hom.ext_iff, OverClass.asOver_hom, PreservesPullbacksOfInclusions.preservesPullbackInr', Discrete.addMonoidalFunctor_δ, Limits.BinaryBicone.ofLimitCone_snd, AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst, Over.η_pullback_left, instInitialDiscreteOfIsConnected, Functor.RightExtension.mk_left, SymmetricCategory.rightDistrib_of_leftDistrib, Functor.toUnder_obj_right, FunctorToTypes.inl_comp_binaryCoproductIso_inv_apply, Limits.MonoCoprod.mono_inj, Over.postAdjunctionRight_unit_app, CostructuredArrow.mapNatIso_functor_obj_left, Under.hom_right_inv_right, Discrete.essentiallySmallOfSmall, Over.id_left, Limits.Types.instSubsingletonTerminalType, ChosenPullbacksAlong.Over.leftUnitor_hom_left, SimplicialObject.Splitting.decomposition_id, Limits.limitConeOfUnique_isLimit_lift, OverPresheafAux.restrictedYonedaObjMap₁_app, MorphismProperty.Under.mk_left, ChosenPullbacksAlong.cartesianMonoidalCategoryFst_pullback_obj, Limits.Cocone.fromStructuredArrow_map_hom, Over.postComp_hom_app_left, Limits.coprod.map_comp_inl_inr_codiag, Limits.CoproductsFromFiniteFiltered.liftToFinset_map_app, Functor.RightExtension.postcomp₁_obj_right, Discrete.id_def', Over.whiskerLeft_left_fst, Limits.hasTerminalChangeDiagram, Discrete.functor_ext_iff, Limits.coprod.braiding_hom, ChosenPullbacksAlong.cartesianMonoidalCategoryFst_mapPullbackAdj_counit_app, SmallObject.πFunctorObj_eq, AlgebraicGeometry.isCompl_opensRange_inl_inr, StructuredArrow.mapNatIso_counitIso_hom_app_right, expComparison_ev, MonoOver.inf_map_app, Discrete.compNatIsoDiscrete_inv_app, CommMon.EquivLaxBraidedFunctorPUnit.counitIso_aux_one, Limits.FormalCoproduct.cofanHomEquiv_symm_apply_f, TwoSquare.costructuredArrowDownwardsPrecomp_map, PreZeroHypercover.inj_sigmaOfIsColimit_f, Limits.MultispanIndex.multicoforkEquivSigmaCofork_inverse_obj_ι_app, CostructuredArrow.ofCommaFstEquivalenceFunctor_obj_left, Limits.MulticospanIndex.multiforkEquivPiFork_inverse_obj_pt, Functor.LeftExtension.postcompose₂_map_left, Limits.BinaryFan.isLimit_iff_isIso_fst, ChosenPullbacksAlong.isoInv_mapPullbackAdj_counit_app_left, Over.post_obj, Functor.LeftExtension.postcomp₁_obj_right_obj, IsGrothendieckAbelian.GabrielPopescuAux.kernel_ι_d_comp_d, Over.ConstructProducts.conesEquiv_functor, MonoOver.mkArrowIso_inv_hom_left, effectiveEpi_desc_iff_effectiveEpiFamily, AlgebraicGeometry.instIsOpenImmersionSigmaSpec, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_ι_isOpenImmersion_aux, Limits.Cone.equivCostructuredArrow_counitIso, MonoidalCategory.DayConvolutionUnit.leftUnitorCorepresentingIso_inv_app_app, Comma.toPUnitIdEquiv_inverse_obj_right_as, StructuredArrow.toUnder_obj_hom, Functor.RightExtension.postcompose₂_obj_hom_app, ObjectProperty.ColimitOfShape.toCostructuredArrow_map, ModuleCat.HasLimit.productLimitCone_cone_pt_isModule, MorphismProperty.Over.instPreservesFiniteLimitsTopOverForget, FundamentalGroupoidFunctor.preservesProduct, Functor.RightExtension.precomp_obj_hom_app, NonPreadditiveAbelian.lift_map_assoc, CostructuredArrow.mapNatIso_functor_obj_right, StructuredArrow.commaMapEquivalenceCounitIso_hom_app_right_right, Pseudofunctor.isPrestackFor_iff, algebraEquivUnder_counitIso, CostructuredArrow.ofCommaFstEquivalenceFunctor_map_left, Functor.RightExtension.precomp_map_right, ChosenPullbacksAlong.Over.lift_left, prod.leftUnitorEquivalence_unitIso, Limits.instReflectsColimitsOfShapeDiscreteOfReflectsFiniteCoproductsOfFinite, Under.postCongr_hom_app_right, Over.mapId_hom_app_left, prod.leftUnitor_isEquivalence, simplyConnectedSpace_iff, Bicategory.LeftLift.ofIdComp_hom, AlgebraicGeometry.Scheme.Hom.toNormalization_inr_normalizationCoprodIso_hom_assoc, WithTerminal.commaFromOver_map_right, CommRingCat.coproductCoconeIsColimit_desc, MonoOver.w_assoc, CostructuredArrow.map_map_left, MorphismProperty.over_iff, CostructuredArrow.map₂_map_left, Limits.prod.diag_map_fst_snd_comp_assoc, MorphismProperty.Under.forget_comp_forget_map, Limits.coprod.map_comp_inl_inr_codiag_assoc, Over.isPullback_of_binaryFan_isLimit, StructuredArrow.ofCommaSndEquivalenceInverse_map_right_right, Pseudofunctor.IsStackFor.isEquivalence, Limits.biprod.conePointUniqueUpToIso_inv, Classifier.SubobjectRepresentableBy.iso_inv_hom_left_comp_assoc, AlgebraicGeometry.Scheme.Cover.pullbackCoverOverProp_X, Limits.BinaryFan.braiding_inv_fst, MorphismProperty.instFullOverTopOverForget, CostructuredArrow.ofCommaFstEquivalenceFunctor_obj_right, Limits.coproductUniqueIso_inv, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand, Limits.BinaryFan.assoc_snd, Bicategory.LeftLift.whiskerOfIdCompIsoSelf_hom_right, Over.whiskerLeft_left_fst_assoc, Functor.RightExtension.postcompose₂ObjMkIso_inv_left_app, Limits.Pi.cone_pt, AlgebraicGeometry.LocallyRingedSpace.instHasColimitsOfShapeDiscrete, TopCat.isEmbedding_pullback_to_prod, CostructuredArrow.preEquivalence.inverse_obj_left_left, MonoOver.pullback_obj_left, CostructuredArrow.ofCostructuredArrowProjEquivalence.functor_obj_left_left, Limits.MultispanIndex.toSigmaCoforkFunctor_obj, Under.pushout_obj, Limits.fst_of_isColimit, MorphismProperty.Over.w, Limits.hasInitialChangeDiagram, Functor.RightExtension.postcompose₂_map_left_app, GradedObject.CofanMapObjFun.ιMapObj_iso_inv_assoc, Limits.CoproductDisjoint.nonempty_isInitial_of_ne, CartesianMonoidalCategory.ofChosenFiniteProducts.pentagon, Over.coprodObj_obj, AlgebraicGeometry.isClosedImmersion_equalizer_ι_left, Limits.FormalCoproduct.instPreservesColimitsOfShapeDiscreteObjFunctorEval, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.trans_app_left, StructuredArrow.mk_right, Limits.CompleteLattice.prod_eq_inf, ChosenPullbacksAlong.Over.associator_inv_left_fst_fst, Functor.RightExtension.postcompose₂_obj_left_obj, ChosenPullbacksAlong.cartesianMonoidalCategorySnd_pullback_obj, CostructuredArrow.ofDiagEquivalence.functor_obj_left_right_as, StructuredArrow.preEquivalenceInverse_map_right_right, isConnected_iff_initial_of_unique, ModuleCat.HasLimit.productLimitCone_cone_pt_carrier, CostructuredArrow.IsUniversal.fac, Subfunctor.equivalenceMonoOver_unitIso, instHasLimitsOfShapeDiscreteInd, Mon.EquivLaxMonoidalFunctorPUnit.counitIso_hom_app_hom, Over.prodComparisonIso_pullback_Spec_inv_left_fst_fst', AlgebraicGeometry.coprodSpec_inl_assoc, PreGaloisCategory.has_decomp_quotients, Limits.MulticospanIndex.sndPiMapOfIsLimit_proj_assoc, Mon.EquivLaxMonoidalFunctorPUnit.counitIsoAux_hom, Mon.EquivLaxMonoidalFunctorPUnit.counitIsoAux_IsMon_Hom, Limits.coprod.rightUnitor_inv, FreeMonoidalCategory.normalizeIsoAux_hom_app, AlgebraicGeometry.Scheme.instLocallyCoverDenseOverTopMorphismPropertyOverForgetOverGrothendieckTopology, Mon.EquivLaxMonoidalFunctorPUnit.laxMonoidalToMon_map, Limits.BinaryFan.braiding_inv_snd_assoc, MonoOver.subobjectMk_le_mk_of_hom, FunctorToTypes.binaryProductIso_inv_comp_fst_apply, Limits.Bicone.ofLimitCone_π, Limits.ι_comp_sigmaObjIso_inv_assoc, Comma.equivProd_unitIso_inv_app_right, Over.whiskerLeft_left_snd_assoc, HomotopicalAlgebra.Cylinder.instIsVeryGoodOfFactorizationData, Limits.Fan.IsLimit.lift_proj, Limits.IsColimit.pushoutCoconeEquivBinaryCofanFunctor_desc_right, preservesBinaryCoproducts_of_preservesInitial_and_pushouts, Limits.prod.pentagon, Limits.MultispanIndex.parallelPairDiagramOfIsColimit_map, TopCat.Presheaf.isSheaf_unit, CostructuredArrow.grothendieckProj_obj, Discrete.addMonoidalFunctor_μ, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap_comp, Over.equivalenceOfIsTerminal_unitIso, TwoSquare.EquivalenceJ.functor_obj, AlgebraicGeometry.sigmaOpenCover_I₀, IsGrothendieckAbelian.generatingMonomorphisms.top_mem_range, Limits.Types.coproductIso_mk_comp_inv_apply, TopologicalSpace.Opens.coe_overEquivalence_inverse_obj_left, Comma.toPUnitIdEquiv_inverse_obj_left, Over.iteratedSliceEquiv_counitIso, Limits.colimitQuotientCoproduct_epi, StructuredArrow.mapIso_unitIso_hom_app_right, AlgebraicGeometry.instPreservesColimitsOfShapeOppositeCommRingCatSchemeDiscretePEmptySpec, piEquivalenceFunctorDiscrete_inverse_map, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand₀_assoc, CategoryOfElements.costructuredArrowULiftYonedaEquivalence_inverse_obj, Limits.CoproductDisjoint.mono_inj, StructuredArrow.w, Functor.RightExtension.postcompose₂_map_right, Over.leftUnitor_inv_left_fst_assoc, Limits.Cone.fromCostructuredArrow_obj_pt, whiskerRight_coprod_inl_rightDistrib_inv_assoc, commAlgCatEquivUnder_inverse_map, Functor.RightExtension.IsPointwiseRightKanExtensionAt.isoLimit_inv_π_assoc, ChosenPullbacksAlong.Over.whiskerLeft_left, IsPushout.of_is_coproduct, Functor.preservesBinaryProducts_of_preservesKernels, Under.forget_obj, FunctorToTypes.inl_comp_binaryCoproductIso_inv, Sieve.forallYonedaIsSheaf_iff_colimit, Discrete.productEquiv_inverse_obj_as, WithTerminal.liftFromOver_obj_map, Functor.relativelyRepresentable.diag_of_map_from_obj, Functor.LeftExtension.precomp_map_left, Abelian.Pseudoelement.pseudoZero_aux, StructuredArrow.commaMapEquivalenceUnitIso_hom_app_right_right, AlgebraicGeometry.ι_sigmaSpec_assoc, CostructuredArrow.mapNatIso_counitIso_hom_app_left, StructuredArrow.pre_map_right, Localization.HasProductsOfShapeAux.inverts, TopCat.prodIsoProd_hom_apply, Limits.preservesBinaryProduct_of_preservesBinaryBiproduct, Limits.Types.binaryProductIso_inv_comp_fst_apply, AlgebraicGeometry.isPullback_inr_inr_coprodMap, StructuredArrow.w_prod_fst_assoc, WithInitial.coconeEquiv_inverse_map_hom_right, Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.comp_homEquiv_symm, coprod_inr_leftDistrib_hom_assoc, Functor.LeftExtension.precomp₂_obj_right, SimplicialObject.Splitting.ι_desc, Discrete.monoidal_rightUnitor, StructuredArrow.mapIso_functor_obj_hom, StructuredArrow.mapNatIso_unitIso_inv_app_right, Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_ι_app, Functor.ranObjObjIsoLimit_hom_π, ChosenPullbacksAlong.cartesianMonoidalCategoryToUnit_mapPullbackAdj_counit_app, WithTerminal.isLimitEquiv_apply_lift_left, subterminalsEquivMonoOverTerminal_unitIso, overToCoalgebra_obj_A, Alexandrov.projSup_obj, Limits.Cofan.cofanTypes_pt, Limits.ReflectsFiniteCoproducts.reflects, AB4OfSize.ofShape, Discrete.addMonoidalFunctor_ε, underToAlgebra_map_f, Limits.Bicone.ofColimitCocone_π, Over.postEquiv_unitIso, Functor.LeftExtension.coconeAt_pt, TopCat.prodIsoProd_hom_fst, SimplicialObject.Splitting.πSummand_comp_cofan_inj_id_comp_PInfty_eq_PInfty_assoc, Limits.preservesProductsOfShape_of_preservesBiproductsOfShape, AlgebraicGeometry.coprodSpec_inl, MorphismProperty.CostructuredArrow.Hom.ext_iff, AddCommGrpCat.HasLimit.lift_hom_apply, Profinite.Extend.cocone_ι_app, NonPreadditiveAbelian.mono_r, Limits.Fan.IsLimit.lift_proj_assoc, Limits.BinaryCofan.ι_app_left, Limits.ProductsFromFiniteCofiltered.finiteSubproductsCone_π_app, Pseudofunctor.IsStackFor.essSurj, Limits.BinaryBicone.toCocone_ι_app_left, Limits.coprod.rightUnitor_hom, Over.associator_inv_left_fst_snd_assoc, ChosenPullbacksAlong.cartesianMonoidalCategorySnd_mapPullbackAdj_counit_app, Limits.Multifork.ofPiFork_π_app_right, Limits.CofanTypes.sigma_ι_fst, Functor.LeftExtension.postcomp₁_map_left, Functor.equiv_functor_map, Functor.RightExtension.coneAtFunctor_map_hom, Limits.HasFiniteProducts.out, HomotopicalAlgebra.AttachCells.cell_def_assoc, MonoOver.forget_obj_hom, Limits.Types.productIso_inv_comp_π_apply, Limits.Types.productIso_inv_comp_π, Classifier.SubobjectRepresentableBy.iso_inv_left_comp, Over.rightUnitor_hom_left, AlgebraicGeometry.Scheme.smallGrothendieckTopology_eq_toGrothendieck_smallPretopology, MorphismProperty.Over.pullback_map_left, whiskerRight_coprod_inr_rightDistrib_inv, Limits.Multifork.ofPiFork_π_app_left, Functor.final_of_isFiltered_of_pUnit, FunctorToTypes.inr_comp_binaryCoproductIso_inv, CatEnriched.id_eq, monoidalOfHasFiniteCoproducts.leftUnitor_hom, Over.ConstructProducts.conesEquivFunctor_obj_pt, Limits.ProductsFromFiniteCofiltered.liftToFinsetLimitCone_cone_π_app, toOver_map_left, Discrete.addMonoidalFunctorComp_isMonoidal, Over.sections_map, Functor.toUnder_map_right, ChosenPullbacksAlong.fst'_left, Functor.RightExtension.coneAtWhiskerRightIso_hom_hom, Bicategory.LeftLift.whiskerIdCancel_right, CostructuredArrow.ofCommaFstEquivalenceInverse_obj_left_right, CartesianMonoidalCategory.prodComparisonIso_id, MonoOver.commSqOfHasStrongEpiMonoFactorisation, MonoidalCategory.DayConvolutionUnit.leftUnitorCorepresentingIso_hom_app_app, rightAdjointOfCostructuredArrowTerminalsAux_symm_apply, StructuredArrow.mapIso_inverse_map_right, Over.μ_pullback_left_snd, Limits.BinaryCofan.IsColimit.desc'_coe, ModuleCat.HasLimit.lift_hom_apply, Under.mapPushoutAdj_counit_app, FunctorToTypes.binaryProductCone_π_app, Over.iteratedSliceBackward_obj, ModuleCat.binaryProductLimitCone_isLimit_lift, subterminalsEquivMonoOverTerminal_counitIso, CostructuredArrow.homMk'_comp, preservesFinOfPreservesBinaryAndTerminal, Limits.Types.binaryCoproductCocone_pt, TopCat.prodIsoProd_inv_snd, Limits.MulticospanIndex.ofPiForkFunctor_obj, isCoseparator_iff_of_isLimit_fan, TwoSquare.structuredArrowDownwards_map, Discrete.sumEquiv_inverse_obj, Limits.Multicofork.sigma_condition, CostructuredArrow.ofDiagEquivalence.inverse_map_left, ChosenPullbacksAlong.Over.toUnit_left, WithTerminal.coneEquiv_unitIso_inv_app_hom_left, Bicategory.LeftExtension.whiskerOfCompIdIsoSelf_hom_right, Classifier.SubobjectRepresentableBy.iso_inv_left_π, Localization.structuredArrowEquiv_symm_apply, Over.map_obj_left, Over.epi_left_of_epi, AlgebraicGeometry.instIsLocallyDirectedCompSchemeOverOverTopMorphismPropertyForgetForgetForget, AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso_inv_coprodDesc_fromNormalization_assoc, TopCat.piIsoPi_inv_π, Functor.LeftExtension.mk_right, Limits.Bicone.toCone_π_app, SimplicialObject.Splitting.cofan_inj_πSummand_eq_zero_assoc, Limits.MultispanIndex.multicoforkEquivSigmaCoforkOfIsColimit_inverse, CostructuredArrow.mapCompιCompGrothendieckProj_hom_app, instIsFilteredDiscretePUnit, Limits.Sigma.map_isIso, Sieve.overEquiv_symm_top, Limits.coprod.triangle, isConnected_iff_final_of_unique, Limits.BinaryFan.leftUnitor_hom, Limits.Fan.nonempty_isLimit_iff_isIso_piLift, Functor.RightExtension.IsPointwiseRightKanExtensionAt.isoLimit_hom_π, StructuredArrow.ofCommaSndEquivalenceFunctor_obj_left, symmetricOfHasFiniteProducts_braiding_inv, StructuredArrow.ofDiagEquivalence.functor_obj_hom, AlgebraicGeometry.sigmaOpenCover_f, FunctorToTypes.binaryCoproductCocone_ι_app, Limits.Sigma.cocone_ι, Limits.Types.pi_lift_π_apply, Bicategory.LeftLift.IsKan.uniqueUpToIso_hom_right, TwoSquare.structuredArrowDownwards_obj, Limits.hasTerminalChangeUniverse, Presieve.piComparison_fac, CommRingCat.Under.equalizerFork_ι, CostructuredArrow.eqToHom_left, CostructuredArrow.mapNatIso_inverse_obj_right, IsSifted.colim_preservesLimitsOfShape_pempty_of_isSifted, Discrete.addMonoidalFunctor_η, Limits.CoproductsFromFiniteFiltered.liftToFinsetColimitCocone_cocone_pt, Limits.BinaryBicone.ofLimitCone_fst, Limits.IsLimit.assoc_lift, Limits.PreservesInitial.of_iso_comparison, Under.postAdjunctionLeft_counit_app, Under.inv_right_hom_right, NatTrans.Coequifibered.of_discrete, Limits.FormalCoproduct.instPreservesColimitDiscreteFunctorObjFunctorEval, CechNerveTerminalFrom.wideCospan.limitIsoPi_inv_comp_pi_assoc, CommMon.EquivLaxBraidedFunctorPUnit.unitIso_inv_app_hom_hom_app, monoidalOfHasFiniteCoproducts.tensorHom, CostructuredArrow.mapNatIso_functor_map_left, Limits.MulticospanIndex.parallelPairDiagramOfIsLimit_obj, Over.ConstructProducts.conesEquivUnitIso_hom_app_hom, instHomIsOverLeftDiscretePUnit, hoFunctor.instIsIsoCatProdComparisonSSetHoFunctorNerve, Over.braiding_hom_left, MonoOver.instMonoHomDiscretePUnitObjOverForget, ChosenPullbacksAlong.Over.associator_inv_left_snd, ChosenPullbacksAlong.isoInv_pullback_obj_hom, WithInitial.liftFromUnder_map_app, Discrete.monoidalFunctor_obj, CostructuredArrow.ιCompGrothendieckPrecompFunctorToCommaCompFst_hom_app, ChosenPullbacksAlong.iso_mapPullbackAdj_counit_app, MorphismProperty.Over.pullback_obj_hom, FinitaryPreExtensive.isUniversal_finiteCoproducts, Limits.BinaryBicone.ofColimitCocone_snd, Comma.equivProd_inverse_obj_left, StructuredArrow.projectSubobject_factors, Functor.leftExtensionEquivalenceOfIso₁_unitIso_hom_app_right_app, countableCategoryDiscreteOfCountable, Limits.ι_colimitPointwiseProductToProductColimit_π_assoc, ChosenPullbacksAlong.Over.whiskerLeft_left_snd, ChosenPullbacksAlong.Over.associator_hom_left_snd_snd, overToCoalgebra_obj_a, Limits.colimitCoconeOfUnique_cocone_pt, MorphismProperty.Over.mapComp_inv_app_left, ChosenPullbacksAlong.isoInv_pullback_obj_right_as, FunctorToTypes.mem_fromOverSubfunctor_iff, CostructuredArrow.hom_ext_iff, HomotopicalAlgebra.AttachCells.isPushout, hoFunctor.preservesBinaryProduct, Functor.preservesTerminalObject_of_preservesZeroMorphisms, Limits.MultispanIndex.toSigmaCoforkFunctor_map_hom, Over.μ_pullback_left_fst_fst, Discrete.opposite_inverse_obj, StructuredArrow.ofDiagEquivalence.inverse_obj_left_as, Under.hom_right_inv_right_assoc, Over.starPullbackIsoStar_inv_app_left, Limits.Types.coproductIso_ι_comp_hom_apply, CommRingCat.piFan_pt, Over.iteratedSliceForward_map, Limits.PreservesLimitPair.of_iso_prod_comparison, TopCat.prodIsoProd_hom_fst_assoc, MorphismProperty.underObj_iff, Limits.MonoCoprod.mono_binaryCofanSum_inr, MonoOver.inf_obj, Presheaf.tautologicalCocone_ι_app, CostructuredArrow.pre_map_right, Limits.hasZeroObject_pUnit, AlgebraicGeometry.Scheme.Cover.pullbackCoverOver'_f, StructuredArrow.mk_hom_eq_self, StructuredArrow.mapNatIso_functor_obj_left, Limits.braid_natural, StructuredArrow.commaMapEquivalenceUnitIso_hom_app_right_left, MorphismProperty.Under.w_assoc, MonoOver.top_left, Over.tensorUnit_left, ObjectProperty.instIsClosedUnderLimitsOfShapeDiscretePEmptyOfContainsZeroOfIsClosedUnderIsomorphisms, Presheaf.tautologicalCocone'_ι_app, Comma.toPUnitIdEquiv_counitIso_inv_app, MorphismProperty.instFullCostructuredArrowTopOverToOver, StructuredArrow.preEquivalenceFunctor_map_right, toOverIteratedSliceForwardIsoPullback_inv_app_left, Over.hom_left_inv_left_assoc, SmallObject.functorObj_comm_assoc, CategoryOfElements.costructuredArrowYonedaEquivalenceInverseπ_inv_app, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_ι_isOpenEmbedding, AlgebraicGeometry.IsImmersion.instLiftSchemeId, StructuredArrow.preEquivalenceInverse_obj_hom_right, Limits.Bicone.ofColimitCocone_pt, prod.leftUnitor_obj, AlgebraicGeometry.Scheme.mem_smallGrothendieckTopology, ChosenPullbacksAlong.iso_pullback_map, Under.opEquivOpOver_inverse_obj, Limits.opProductIsoCoproduct'_inv_comp_lift, IsSifted.colim_preservesTerminal_of_isSifted, Over.ConstructProducts.conesEquivInverse_obj, AlgebraicGeometry.nonempty_isColimit_binaryCofanMk_of_isCompl, ModuleCat.HasLimit.productLimitCone_cone_pt_isAddCommGroup, Limits.FormalCoproduct.coproductIsoSelf_inv_φ, Limits.BinaryFan.leftUnitor_inv, Limits.piObjIso_inv_comp_π_assoc, rightAdjoint_preservesInitial_of_coreflective, typeToCat_obj, AlgebraicGeometry.Scheme.Hom.inl_toNormalization_normalizationCoprodIso_inv_assoc, Types.monoOverEquivalenceSet_unitIso, Discrete.monoidalFunctor_μ, Functor.ranObjObjIsoLimit_inv_π, Limits.pair_obj_left, leftDistrib_hom, TwoSquare.EquivalenceJ.functor_map, Under.UnderMorphism.ext_iff, CostructuredArrow.costructuredArrowToOverEquivalence.inverse_obj, Over.iteratedSliceEquiv_inverse, StructuredArrow.mapIso_inverse_obj_left, Over.coe_hom, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_obj_obj_left, MorphismProperty.instHasPullbackSndHomDiscretePUnitOfHasPullbacksAlongOfIsStableUnderBaseChangeAlong, AlgebraicGeometry.instIsIsoSchemeCoprodSpec, Limits.coprod.associator_hom, AlgebraicGeometry.Scheme.kerFunctor_obj, HomotopicalAlgebra.Cylinder.instIsVeryGoodSymmOfRespectsIsoCofibrations, ChosenPullbacksAlong.pullbackIsoOverPullback_hom_app_comp_fst_assoc, Limits.Types.pi_map_π_apply', Over.forgetCocone_ι_app, Limits.MultispanIndex.multicoforkEquivSigmaCofork_inverse_map_hom, Limits.Pi.isoLimit_hom_π, Monoidal.rightUnitor_inv, StructuredArrow.commaMapEquivalenceUnitIso_inv_app_right_left, Discrete.equivOfEquivalence_symm_apply, Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_inv, ModuleCat.binaryProductLimitCone_cone_pt, Limits.preservesColimitsOfShape_pempty_of_preservesInitial, StructuredArrow.mapIso_functor_map_right, Discrete.addMonoidal_associator, prod.rightInverseUnitor_obj, Under.costar_map_left, CommGrpCat.binaryProductLimitCone_isLimit_lift, NonPreadditiveAbelian.diag_σ_assoc, Over.whiskerRight_left_fst_assoc, Limits.Types.binaryProductIso_inv_comp_snd_apply, Limits.preservesBinaryProducts_of_preservesBinaryBiproducts, StructuredArrow.pre_obj_hom, AlgebraicGeometry.isEmpty_pullback_sigmaι_of_ne, CostructuredArrow.mapIso_functor_obj_right, SmallObject.FunctorObjIndex.comm_assoc, instHasColimitsOfShapeDiscreteIndOfFinite, Limits.preservesTerminal_of_isIso, Bicategory.LanLift.CommuteWith.lanLiftCompIsoWhisker_inv_right, Discrete.functor_map, Discrete.sumEquiv_functor_map, Limits.MulticospanIndex.multiforkEquivPiForkOfIsLimit_unitIso, Over.tensorHom_left_snd, Bicategory.LeftExtension.whiskerHom_right, Limits.Cone.overPost_π_app, Under.opEquivOpOver_counitIso, StructuredArrow.toUnder_map_left, StructuredArrow.w_prod_snd_assoc, AlgebraicGeometry.IsIntegralHom.instDescScheme, CostructuredArrow.preEquivalence.inverse_obj_hom_left, LightCondensed.instCountableAB4StarLightCondMod, StructuredArrow.prodInverse_obj, MonoidalCategory.DayConvolutionUnit.rightUnitorCorepresentingIso_inv_app_app, NonPreadditiveAbelian.sub_def, Limits.instHasColimitOppositeDiscreteOpFunctor, Comma.toIdPUnitEquiv_counitIso_inv_app, Limits.proj_comp_opProductIsoCoproduct'_hom, CostructuredArrow.preEquivalence.functor_map_left, StructuredArrow.preEquivalence_counitIso, Limits.IsLimit.binaryFanSwap_lift, Functor.RightExtension.postcomp₁_obj_hom_app, Limits.Multicofork.sigma_condition_assoc, StructuredArrow.ext_iff, Over.forget_map, Subobject.representative_coe, Limits.Multifork.pi_condition_assoc, Functor.ι_leftKanExtensionObjIsoColimit_hom, Limits.SequentialProduct.cone_π_app, Limits.MulticospanIndex.ofPiForkFunctor_map_hom, WithInitial.liftFromUnder_obj_map, Limits.PreservesFiniteCoproducts.preserves, CostructuredArrow.ofCostructuredArrowProjEquivalence.inverse_obj_left_right_as, WithTerminal.liftFromOver_map_app, typeToCat_map, StructuredArrow.preEquivalenceInverse_obj_left_as, Limits.Cofan.nonempty_isColimit_iff_isIso_sigmaDesc, TopCat.prodIsoProd_inv_fst_apply, AlgebraicTopology.DoldKan.Γ₀.Obj.mapMono_on_summand_id_assoc, Sieve.overEquiv_generate, Presieve.Extensive.arrows_nonempty_isColimit, Discrete.addMonoidal_tensorUnit_as, rightDistrib_hom, WithTerminal.coneEquiv_inverse_map_hom_left, Limits.BinaryBicone.toCocone_pt, FundamentalGroupoid.punitEquivDiscretePUnit_counitIso, TopCat.sigmaCofan_pt, Bicategory.LeftLift.ofIdComp_left_as, AlgebraicGeometry.Scheme.Hom.toNormalization_inl_normalizationCoprodIso_hom, Bicategory.LeftExtension.whiskerOfCompIdIsoSelf_inv_right, Subobject.inf_eq_map_pullback, Over.postMap_app, Limits.PreservesProduct.of_iso_comparison, AlgebraicGeometry.Scheme.Hom.inr_toNormalization_normalizationCoprodIso_inv_assoc, StructuredArrow.preEquivalenceFunctor_obj_right, AlgebraicGeometry.Scheme.instIsClosedImmersionLiftIdOfIsSeparated, instIsCofilteredDiscretePUnit, Limits.Cofan.IsColimit.fac_assoc, Under.mkIdInitial_to_right, CostructuredArrow.preEquivalence.inverse_obj_right_as, CostructuredArrow.grothendieckPrecompFunctorToComma_map_right, coprod_inl_rightDistrib_hom_assoc, HomotopicalAlgebra.Cylinder.instIsGoodTrans, Over.ConstructProducts.conesEquivInverseObj_π_app, NonPreadditiveAbelian.lift_map, Enriched.HasConicalProducts.hasConicalLimitsOfShape, TopologicalSpace.Opens.overEquivalence_inverse_obj_right_as, Limits.pointwiseProductCompEvaluation_hom_app, StructuredArrow.proj_obj, coprodComparison_tensorLeft_braiding_hom, Under.costar_obj_hom, Functor.LeftExtension.coconeAt_ι_app, Bicategory.LeftLift.IsKan.uniqueUpToIso_inv_right, Comma.equivProd_functor_map, Presheaf.isLimit_iff_isSheafFor, ChosenPullbacksAlong.pullbackIsoOverPullback_inv_app_comp_fst_assoc, FunctorToTypes.binaryProductCone_pt_map, CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_ε, StructuredArrow.ofStructuredArrowProjEquivalence.functor_obj_right_left_as, CartesianMonoidalCategory.preservesLimit_empty_of_isIso_terminalComparison, Over.lift_left, piEquivalenceFunctorDiscrete_functor_map, Limits.CoproductsFromFiniteFiltered.finiteSubcoproductsCocone_ι_app_eq_sum, Limits.prod.diag_map_fst_snd_comp, Limits.FormalCoproduct.ι_comp_coproductIsoCofanPt, Under.postAdjunctionRight_counit_app_right, Limits.pushoutCoconeEquivBinaryCofan_counitIso, Limits.preservesInitial_of_iso, Functor.LeftExtension.mk_left_as, Limits.colimitCoconeOfUnique_cocone_ι, TwoSquare.EquivalenceJ.inverse_obj, AlgebraicGeometry.Scheme.restrictFunctor_obj_left, Limits.Pi.isoLimit_hom_π_assoc, Limits.BinaryFan.assocInv_snd, HomotopicalAlgebra.Precylinder.symm_i, MonoOver.congr_counitIso, CartesianMonoidalCategory.fullSubcategory_tensorProductIsBinaryProduct_lift_hom, StructuredArrow.map_obj_hom, CostructuredArrow.preEquivalence.inverse_map_left_left, Limits.coprod.symmetry'_assoc, essentiallySmall_of_le, SmallObject.functorObj_isPushout, Discrete.equivalence_inverse, Over.opEquivOpUnder_functor_map, CartesianMonoidalCategory.preservesTerminalIso_id, Limits.mapPair_right, Limits.SequentialProduct.functorMap_commSq_succ, Limits.Cofan.IsColimit.fac, Limits.Multifork.ofPiFork_ι, Functor.LeftExtension.precomp₂_map_left, CommMon.EquivLaxBraidedFunctorPUnit.counitIso_inv_app_hom_hom, Limits.HasBiproductsOfShape.colimIsoLim_hom_app, OverPresheafAux.counitForward_counitBackward, AlgebraicGeometry.instMonoObjWalkingSpanCompOverSchemeTopMorphismPropertySpanOverForgetForgetForgetNoneWalkingPairSomeMapInitOfIsOpenImmersionLeftDiscretePUnit, SSet.Truncated.rightExtensionInclusion_hom_app, NonPreadditiveAbelian.lift_σ, ChosenPullbacksAlong.cartesianMonoidalCategoryToUnit_pullback_map, FreeMonoidalCategory.normalizeIsoApp_tensor, StructuredArrow.mapIso_inverse_map_left, NonPreadditiveAbelian.lift_σ_assoc, Limits.MonoCoprod.mono_binaryCofanSum_inl, AlgebraicGeometry.Scheme.restrictFunctor_map_left, Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_laxMonoidalToMon_obj_mul, ChosenPullbacksAlong.pullbackIsoOverPullback_hom_app_comp_fst, TopCat.Presheaf.generateEquivalenceOpensLe_functor'_map, Limits.BinaryFan.assocInv_fst, CostructuredArrow.ιCompGrothendieckProj_hom_app, Limits.opCoproductIsoProduct'_hom_comp_proj_assoc, CostructuredArrow.pre_map_left, Limits.BinaryFan.π_app_left, Mon.EquivLaxMonoidalFunctorPUnit.laxMonoidalToMon_obj, Limits.MultispanIndex.multicoforkEquivSigmaCofork_unitIso_hom_app_hom, Limits.hasLimitsOfShape_discrete, Limits.Sigma.ι_isoColimit_inv_assoc, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.functor_map, Limits.prod.map_swap, Functor.equiv_inverse, HomotopicalAlgebra.weakEquivalences_over_iff, Discrete.productEquiv_functor_obj, StructuredArrow.ofCommaSndEquivalenceInverse_obj_right_hom, Over.ConstructProducts.over_product_of_widePullback, Under.inv_right_hom_right_assoc, CostructuredArrow.ext_iff, LocalizerMorphism.IsRightDerivabilityStructure.Constructor.fromRightResolution_obj, AlgebraicTopology.DoldKan.Γ₀.Obj.map_epi_on_summand_id, Over.mkIdTerminal_from_left, Limits.MultispanIndex.multicoforkEquivSigmaCoforkOfIsColimit_unitIso, ChosenPullbacksAlong.Over.fst_eq_fst', Discrete.monoidalFunctor_η, FundamentalGroupoid.punitEquivDiscretePUnit_inverse, TopCat.sigmaIsoSigma_hom_ι_apply, CostructuredArrow.mapNatIso_functor_map_right, WithTerminal.coneEquiv_functor_obj_pt, ObjectProperty.LimitOfShape.toStructuredArrow_map, Under.opEquivOpOver_functor_map, prod.leftUnitor_map, Limits.diagramIsoPair_inv_app, Over.ConstructProducts.conesEquiv_inverse, CostructuredArrow.mapIso_inverse_obj_hom, CartesianMonoidalCategory.ofChosenFiniteProducts.rightUnitor_naturality, Under.postEquiv_unitIso, CostructuredArrow.projectQuotient_factors, AlgebraicGeometry.IsLocalAtSource.sigmaDesc, Functor.essImage_overPost, TopologicalSpace.Opens.overEquivalence_counitIso_hom_app, Over.fst_left, Over.prodLeftIsoPullback_hom_fst, Limits.Pi.isoLimit_inv_π_assoc, Over.associator_hom_left_fst_assoc, TwoSquare.lanBaseChange_app, IsGrothendieckAbelian.GabrielPopescuAux.ι_d_assoc, Comma.equivProd_unitIso_inv_app_left, Limits.pullbackConeEquivBinaryFan_unitIso, SimplicialObject.Splitting.toKaroubiNondegComplexIsoN₁_hom_f_f, Over.isMonHom_pullbackFst_id_right, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand', Limits.prod.leftUnitor_inv_naturality, Over.pullback_obj_hom, Limits.Multicofork.ofSigmaCofork_ι_app_left, Over.forgetAdjStar_unit_app_left, Discrete.functorComp_inv_app, AddGrpCat.binaryProductLimitCone_cone_pt, HomotopicalAlgebra.instWeakEquivalenceLeftDiscretePUnitOfOver, Pseudofunctor.isPrestackFor_iff_isSheafFor', isSeparator_of_isColimit_cofan, CommRingCat.prodFan_pt, Limits.prod.leftUnitor_hom_naturality, Discrete.isDiscrete, Limits.BinaryFan.assoc_fst, CostructuredArrow.epi_iff_epi_left, StructuredArrow.toCostructuredArrow'_map, Limits.coprod.map_swap_assoc, HomotopicalAlgebra.Cylinder.symm_i_assoc, HomotopicalAlgebra.AttachCells.reindex_isColimit₂, Limits.IndObjectPresentation.yoneda_I, Over.tensorObj_left, FinitaryExtensive.isPullback_initial_to, Limits.CoproductsFromFiniteFiltered.liftToFinset_obj_obj, CostructuredArrow.preEquivalence.inverse_obj_left_right_as, CommMon.EquivLaxBraidedFunctorPUnit.counitIso_aux_mul, AddCommGrpCat.binaryProductLimitCone_cone_π_app_right, HomologicalComplex.instHasColimitDiscreteWalkingPairCompPairEval, LightCondensed.instEpiLightCondModMapNat, Types.monoOverEquivalenceSet_counitIso, PreOneHypercover.forkOfIsColimit_ι_map_inj, FinitaryPreExtensive.universal', StructuredArrow.post_obj, CostructuredArrow.unop_left_comp_underlyingIso_hom_unop, Comma.toIdPUnitEquiv_functor_iso, instIsCocontinuousOverLeftDiscretePUnitIteratedSliceForwardOver, Limits.ProductsFromFiniteCofiltered.liftToFinset_obj_obj, CostructuredArrow.mapIso_inverse_map_right, MonoidalClosed.leftDistrib_inv, AlgebraicGeometry.LocallyRingedSpace.instPreservesColimitsOfShapeSheafedSpaceCommRingCatDiscreteForgetToSheafedSpace, StructuredArrow.toCostructuredArrow'_obj, Abelian.app_hom, StructuredArrow.ofCommaSndEquivalenceInverse_obj_left_as, CostructuredArrow.ofCostructuredArrowProjEquivalence.inverse_obj_left_left, SmallObject.hasColimitsOfShape_discrete, Limits.Cofan.isColimit_cofanTypes_iff, Limits.SequentialProduct.cone_π_app_comp_Pi_π_neg, Functor.leftExtensionEquivalenceOfIso₁_counitIso_hom_app_right_app, Over.prodComparisonIso_pullback_inv_left_fst_snd', Limits.colimitCoconeOfUnique_isColimit_desc, toOverUnitPullback_hom_app_left, Limits.mapPairIso_hom_app, Limits.MultispanIndex.ofSigmaCoforkFunctor_map_hom, Bicategory.LeftExtension.w, prod.rightInverseUnitor_map, FinitaryPreExtensive.isPullback_sigmaDesc, Under.id_right, PreOneHypercover.p₂_sigmaOfIsColimit, mapPair_equifibered, Functor.EssImageSubcategory.lift_def, Limits.CoproductsFromFiniteFiltered.liftToFinsetColimIso_aux, Limits.FormalCoproduct.coproductIsoSelf_hom_φ, Limits.BinaryFan.isLimit_iff_isIso_snd, SimplicialObject.Split.cofan_inj_naturality_symm, Limits.Bicone.ofLimitCone_pt, instIsDenseSubsiteOverLeftDiscretePUnitOverInverseIteratedSliceEquiv, Bicategory.LeftLift.whiskerOfIdCompIsoSelf_inv_right, Over.ε_pullback_left, Over.coprod_map_app, CommMon.EquivLaxBraidedFunctorPUnit.counitIso_hom_app_hom_hom, Functor.ι_leftKanExtensionObjIsoColimit_hom_assoc, AlgebraicGeometry.Scheme.Cover.ColimitGluingData.functor_obj, CostructuredArrow.mapIso_inverse_map_left, CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_map, StructuredArrow.IsUniversal.fac_assoc, StructuredArrow.mapNatIso_inverse_map_left, Presheaf.subsingleton_iff_isSeparatedFor, Limits.Cocone.fromStructuredArrow_obj_ι, CostructuredArrow.map₂_obj_hom, SimplicialObject.Splitting.cofan_inj_comp_app_assoc, ModuleCat.HasLimit.productLimitCone_isLimit_lift, Limits.Cocone.equivStructuredArrow_counitIso, Monoidal.leftUnitor_inv, Limits.FormalCoproduct.cofanHomEquiv_apply_φ, ChosenPullbacksAlong.Over.leftUnitor_inv_left_snd_assoc, ChosenPullbacksAlong.Over.rightUnitor_inv_left_fst_assoc, HomotopicalAlgebra.AttachCells.ofArrowIso_g₁, TopCat.Presheaf.generateEquivalenceOpensLe_inverse'_map, Functor.RightExtension.IsPointwiseRightKanExtensionAt.isoLimit_hom_π_assoc, FunctorToTypes.binaryCoproductCocone_pt_obj, CategoryOfElements.costructuredArrowULiftYonedaEquivalence_inverse_map, hoFunctor.isIso_prodComparison, SmallObject.ι_functorMapTgt_assoc, Discrete.productEquiv_functor_map, CostructuredArrow.map₂_obj_left, MorphismProperty.le_colimitsOfShape_punit, TwoSquare.costructuredArrowRightwards_obj, TopCat.piFan_π_app, Comma.toIdPUnitEquiv_inverse_obj_right, AlgebraicGeometry.Scheme.mem_toGrothendieck_smallPretopology, Limits.Fan.mk_π_app, Limits.Types.isIso_colimitPointwiseProductToProductColimit, MorphismProperty.Over.mapId_hom_app_left, Limits.Cofan.nonempty_isColimit_iff_bijective_fromSigma, Limits.preservesProduct_of_preservesBiproduct, Under.postAdjunctionLeft_unit_app, StructuredArrow.IsUniversal.fac, AlgebraicGeometry.Scheme.restrictFunctor_obj_hom, Limits.MultispanIndex.multicoforkEquivSigmaCofork_unitIso_inv_app_hom, Limits.BinaryFan.mk_pt, instHasExactLimitsOfShapeDiscreteAddCommGrpCat, StructuredArrow.ofDiagEquivalence.functor_obj_left_as, Limits.HasBinaryBiproduct.hasColimit_pair, MorphismProperty.Over.pullbackComp_left_fst_fst, OverPresheafAux.counitAuxAux_hom, Discrete.natIsoFunctor_inv_app, CostructuredArrow.ofCommaFstEquivalenceInverse_obj_hom, StructuredArrow.toUnder_obj_right, Over.associator_hom_left_snd_snd, instIsIsoSSetProdComparisonCatCompNerveFunctorHoFunctorOf, Limits.BinaryBicone.ofLimitCone_pt, Over.associator_inv_left_snd_assoc, Limits.MultispanIndex.parallelPairDiagramOfIsColimit_obj, Limits.preservesLimitsOfShape_pempty_of_preservesTerminal, Over.coprodObj_map, Limits.MultispanIndex.inj_fstSigmaMapOfIsColimit, StructuredArrow.commaMapEquivalenceCounitIso_inv_app_left_right, toOverUnitPullback_inv_app_left, PreOneHypercover.p₂_sigmaOfIsColimit_assoc, piEquivalenceFunctorDiscrete_counitIso, ChosenPullbacksAlong.Over.rightUnitor_inv_left_fst, Under.w_assoc, StructuredArrow.w_assoc, AlgebraicGeometry.coprodMk_inr, StructuredArrow.mapNatIso_functor_map_left, Limits.Types.binaryCoproductIso_inl_comp_hom, ChosenPullbacksAlong.pullbackIsoOverPullback_inv_app_comp_snd_assoc, FunctorToTypes.prod_ext'_iff, Limits.Sigma.cocone_pt, StructuredArrow.map₂_obj_hom, Limits.HasBiproductsOfShape.colimIsoLim_inv_app, StructuredArrow.mapNatIso_counitIso_inv_app_right, HomotopicalAlgebra.PathObject.instIsGoodSymmOfRespectsIsoFibrations, Discrete.productEquiv_inverse_map, Condensed.instAB4CondensedMod, WithInitial.coconeEquiv_unitIso_inv_app_hom_right, CechNerveTerminalFrom.wideCospan.limitIsoPi_hom_comp_pi, AlgebraicGeometry.quasiSeparatedSpace_iff_quasiCompact_prod_lift, CommMon.equivLaxBraidedFunctorPUnit_counitIso, Under.opEquivOpOver_inverse_map, StructuredArrow.commaMapEquivalenceFunctor_obj_right, Limits.Multifork.toPiFork_π_app_zero, SimplicialObject.Splitting.cofan_inj_eq_assoc, Functor.leftExtensionEquivalenceOfIso₁_functor_map_right, Limits.biproduct.conePointUniqueUpToIso_hom, CommRingCat.coproductCocone_ι, AlgebraicGeometry.instHasFiniteCoproductsOverSchemeTopMorphismProperty, Under.forgetCone_π_app, Limits.coprod.symmetry', Over.iteratedSliceForward_obj, Discrete.monoidalFunctor_δ, Limits.Types.binaryProductLimit_lift, Functor.essImage.of_overPost, Limits.CompleteLattice.finite_product_eq_finset_inf, FundamentalGroupoidFunctor.coneDiscreteComp_obj_mapCone, Functor.IsTriangulated.instPreservesLimitsOfShapeDiscreteWalkingPair, TopCat.isEmbedding_prodMap, StructuredArrow.ofStructuredArrowProjEquivalence.inverse_obj_right_hom, Functor.RightExtension.postcompose₂ObjMkIso_hom_left_app, Limits.MultispanIndex.multicoforkEquivSigmaCoforkOfIsColimit_functor, CostructuredArrow.projectQuotient_mk, Limits.CompleteLattice.coprod_eq_sup, Functor.equiv_counitIso, CostructuredArrow.grothendieckPrecompFunctorToComma_obj_hom, AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand'_assoc, discreteEquiv_apply, CostructuredArrow.preEquivalence.functor_obj_left, HomotopicalAlgebra.AttachCells.reindex_isColimit₁, CostructuredArrow.toStructuredArrow_map, Over.whiskerLeft_left_snd, StructuredArrow.ofCommaSndEquivalenceFunctor_obj_hom, Limits.Types.coproductIso_ι_comp_hom, Over.w_assoc, OverPresheafAux.counitForward_val_fst, Discrete.natIso_inv_app, Functor.leftExtensionEquivalenceOfIso₁_unitIso_inv_app_right_app, StructuredArrow.homMk'_mk_id, CostructuredArrow.IsUniversal.fac_assoc, Pseudofunctor.IsPrestackFor.nonempty_fullyFaithful, Alexandrov.projSup_map, FunctorToTypes.binaryProductIso_hom_comp_fst, CompHausLike.instHasColimitsOfShapeDiscreteOfHasExplicitFiniteCoproductsOfFinite, TopCat.Presheaf.SheafConditionEqualizerProducts.res_π_apply, Subfunctor.equivalenceMonoOver_counitIso, Limits.diagonal_pullback_fst, Bicategory.RightLift.w, Over.iteratedSliceForwardNaturalityIso_inv_app, CategoryOfElements.fromCostructuredArrow_obj_fst, StructuredArrow.pre_obj_left, Under.w, ChosenPullbacksAlong.Over.associator_hom_left_fst_assoc, IsPullback.of_is_product, FunctorToTypes.binaryProductIso_inv_comp_fst, Mon.EquivLaxMonoidalFunctorPUnit.unitIso_inv_app_hom_app, equiv_punit_iff_unique, SheafOfModules.Presentation.map_π_eq, CostructuredArrow.costructuredArrowToOverEquivalence.inverse_map, TwoSquare.structuredArrowRightwardsOpEquivalence.functor_obj_left_left_as, CostructuredArrow.mapIso_counitIso_inv_app_left, MorphismProperty.Over.pullbackCongr_hom_app_left_fst_assoc, MonoOver.mono_obj_hom, Over.ConstructProducts.conesEquivFunctor_obj_π_app, monoidalOfHasFiniteProducts.tensorObj, Limits.FintypeCat.nonempty_pi_of_nonempty, Sieve.overEquiv_iff, ChosenPullbacksAlong.Over.tensorHom_left_snd, SimplicialObject.Splitting.cofan_inj_epi_naturality, Limits.prod.associator_naturality, Over.opEquivOpUnder_functor_obj, FunctorToTypes.binaryProductIso_inv_comp_snd, isMonoidalLeftDistrib.of_endofunctors, Limits.MulticospanIndex.multiforkEquivPiFork_unitIso_hom_app_hom, Limits.Types.binaryCoproductIso_inl_comp_inv_apply, HomotopicalAlgebra.fibrations_over_iff, Discrete.natTrans_app, Limits.prod_rightUnitor_inv_naturality, WithTerminal.widePullbackShapeEquiv_functor_obj, StructuredArrow.commaMapEquivalenceCounitIso_inv_app_right_right, Over.ConstructProducts.conesEquivInverse_map_hom, StructuredArrow.mapIso_counitIso_hom_app_right, CommRingCat.Under.equalizer_comp, Functor.costructuredArrowMapCocone_ι_app, Over.iteratedSliceForwardIsoPost_hom_app, Comma.equivProd_functor_obj, Bicategory.RightExtension.w, WithTerminal.coneEquiv_inverse_obj_pt_right_as, MorphismProperty.CostructuredArrow.mk_hom, Over.tensorObj_hom, StructuredArrow.toCostructuredArrow_obj, Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.comp_homEquiv_symm_assoc, toOverUnit_obj_hom, Over.postAdjunctionLeft_counit_app_left, FunctorToTypes.inr_comp_binaryCoproductIso_inv_apply, Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidalObj_ε, CostructuredArrow.preEquivalence.inverse_obj_left_hom, StructuredArrow.ofStructuredArrowProjEquivalence.functor_obj_hom, Limits.pullbackConeEquivBinaryFan_inverse_map_hom, StructuredArrow.hom_ext_iff, StructuredArrow.mapNatIso_inverse_obj_right, StructuredArrow.ofCommaSndEquivalenceInverse_obj_right_right, StructuredArrow.preEquivalenceFunctor_obj_hom, StructuredArrow.mapIso_counitIso_inv_app_right, TopCat.Presheaf.generateEquivalenceOpensLe_inverse, AlgebraicGeometry.Scheme.Hom.inl_toNormalization_normalizationCoprodIso_inv, AlgebraicGeometry.Scheme.Hom.inr_normalizationCoprodIso_hom_fromNormalization, Over.mapComp_inv_app_left, TopCat.piIsoPi_inv_π_assoc, Bicategory.LanLift.CommuteWith.lanLiftCompIsoWhisker_hom_right, Limits.Cofan.cofanTypes_ι, Limits.mapPair_left, HomotopicalAlgebra.AttachCells.hm, Functor.LeftExtension.mk_hom, Under.comp_right, HasPullbacksOfInclusions.hasPullbackInr', Over.mono_left_of_mono, ShiftedHom.opEquiv'_add_symm, MorphismProperty.costructuredArrowObj_iff, MorphismProperty.Over.map_obj_hom, StructuredArrow.commaMapEquivalenceFunctor_obj_hom, CostructuredArrow.IsUniversal.hom_desc, ChosenPullbacksAlong.Over.associator_inv_left_fst_fst_assoc, TopCat.prodIsoProd_inv_snd_apply, MonoOver.isIso_iff_isIso_left, StructuredArrow.pre_obj_right, ChosenPullbacksAlong.Over.associator_hom_left_snd_snd_assoc, Limits.SequentialProduct.functorMap_commSq, Limits.MonoCoprod.binaryCofan_inr, MorphismProperty.instFaithfulUnderTopUnderForget, FreeMonoidalCategory.tensorFunc_obj_map, Limits.ProductsFromFiniteCofiltered.liftToFinsetObj_map, MonoOver.instIsIsoLeftDiscretePUnitHomFullSubcategoryOverIsMono, Functor.RightExtension.precomp_obj_left, WithTerminal.coneEquiv_functor_map_hom, Bicategory.LeftLift.w, Functor.leftExtensionEquivalenceOfIso₁_functor_obj_right, ChosenPullbacksAlong.Over.tensorHom_left, Presheaf.isSheaf_iff_isLimit_pretopology, CostructuredArrow.ofCommaFstEquivalenceInverse_map_left_right, Functor.structuredArrowMapCone_π_app, CostructuredArrow.mapIso_inverse_obj_left, prod.leftUnitorEquivalence_functor, Limits.Multifork.pi_condition, OverPresheafAux.counitForward_naturality₂, CommRingCat.pushout_inr_tensorProdObjIsoPushoutObj_inv_right, Localization.structuredArrowEquiv_apply, FundamentalGroupoid.punitEquivDiscretePUnit_functor, whiskerLeft_coprod_inr_leftDistrib_inv_assoc, Limits.Bicone.ι_of_isLimit

CategoryTheory.Discrete

Definitions

Name	Category	Theorems
addMonoidal 📖	CompOp	46 math math: CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon_map, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.isMonHom_counitIsoAux, CategoryTheory.Mon.equivLaxMonoidalFunctorPUnit_inverse, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.unitIso_hom_app_hom_hom_app, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_map_hom_app, CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_unitIso, addMonoidal_tensorObj_as, CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_functor, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided_map_hom_hom_app, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraided_obj, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_laxMonoidalToMon_obj_one, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.laxBraidedToCommMon_obj, CategoryTheory.Mon.equivLaxMonoidalFunctorPUnit_counitIso, CategoryTheory.Mon.equivLaxMonoidalFunctorPUnit_unitIso, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_obj, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.unitIso_hom_app_hom_app, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.counitIso_inv_app_hom, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidalObj_μ, CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_inverse, addMonoidal_rightUnitor, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.counitIsoAux_inv, CategoryTheory.Mon.equivLaxMonoidalFunctorPUnit_functor, addMonoidal_leftUnitor, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_μ, addMonoidalFunctor_δ, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_aux_one, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.counitIso_hom_app_hom, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.counitIsoAux_hom, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.counitIsoAux_IsMon_Hom, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.laxMonoidalToMon_map, addMonoidalFunctor_μ, addMonoidalFunctor_ε, addMonoidalFunctorComp_isMonoidal, addMonoidalFunctor_η, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.unitIso_inv_app_hom_hom_app, addMonoidal_associator, addMonoidal_tensorUnit_as, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.commMonToLaxBraidedObj_ε, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_inv_app_hom_hom, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidal_laxMonoidalToMon_obj_mul, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.laxMonoidalToMon_obj, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_aux_mul, CategoryTheory.CommMon.EquivLaxBraidedFunctorPUnit.counitIso_hom_app_hom_hom, CategoryTheory.CommMon.equivLaxBraidedFunctorPUnit_counitIso, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.unitIso_inv_app_hom_app, CategoryTheory.Mon.EquivLaxMonoidalFunctorPUnit.monToLaxMonoidalObj_ε
addMonoidalFunctor 📖	CompOp	7 math math: addMonoidalFunctor_obj, addMonoidalFunctor_δ, addMonoidalFunctor_μ, addMonoidalFunctor_ε, addMonoidalFunctorComp_isMonoidal, addMonoidalFunctor_η, CategoryTheory.ShiftedHom.opEquiv'_add_symm
addMonoidalFunctorComp 📖	CompOp	1 math math: addMonoidalFunctorComp_isMonoidal
addMonoidalFunctorMonoidal 📖	CompOp	5 math math: addMonoidalFunctor_δ, addMonoidalFunctor_μ, addMonoidalFunctor_ε, addMonoidalFunctorComp_isMonoidal, addMonoidalFunctor_η
monoidal 📖	CompOp	10 math math: monoidal_associator, monoidal_tensorObj_as, monoidal_tensorUnit_as, monoidalFunctorComp_isMonoidal, monoidal_leftUnitor, monoidalFunctor_ε, monoidal_rightUnitor, monoidalFunctor_μ, monoidalFunctor_η, monoidalFunctor_δ
monoidalFunctor 📖	CompOp	6 math math: monoidalFunctorComp_isMonoidal, monoidalFunctor_ε, monoidalFunctor_obj, monoidalFunctor_μ, monoidalFunctor_η, monoidalFunctor_δ
monoidalFunctorComp 📖	CompOp	1 math math: monoidalFunctorComp_isMonoidal
monoidalFunctorMonoidal 📖	CompOp	5 math math: monoidalFunctorComp_isMonoidal, monoidalFunctor_ε, monoidalFunctor_μ, monoidalFunctor_η, monoidalFunctor_δ

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
addMonoidalFunctorComp_isMonoidal 📖	mathematical	—	CategoryTheory.NatTrans.IsMonoidal CategoryTheory.Discrete CategoryTheory.discreteCategory addMonoidal CategoryTheory.Functor.comp addMonoidalFunctor AddMonoidHom.comp AddZeroClass.toAddZero AddMonoid.toAddZeroClass CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category addMonoidalFunctorComp CategoryTheory.Functor.LaxMonoidal.comp CategoryTheory.Functor.Monoidal.toLaxMonoidal addMonoidalFunctorMonoidal	—	AddMonoidHom.map_zero CategoryTheory.eqToHom_map CategoryTheory.eqToHom_trans CategoryTheory.Category.comp_id AddMonoidHom.map_add CategoryTheory.MonoidalCategory.tensorHom_id CategoryTheory.MonoidalCategory.id_whiskerRight CategoryTheory.Category.id_comp
addMonoidalFunctor_obj 📖	mathematical	—	CategoryTheory.Functor.obj CategoryTheory.Discrete CategoryTheory.discreteCategory addMonoidalFunctor DFunLike.coe AddMonoidHom AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddMonoidHom.instFunLike	—	—
addMonoidalFunctor_δ 📖	mathematical	—	CategoryTheory.Functor.OplaxMonoidal.δ CategoryTheory.Discrete CategoryTheory.discreteCategory addMonoidal addMonoidalFunctor CategoryTheory.Functor.Monoidal.toOplaxMonoidal addMonoidalFunctorMonoidal eqToHom DFunLike.coe AddMonoidHom AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddMonoidHom.instFunLike AddZero.toAdd as AddMonoidHom.map_add	—	—
addMonoidalFunctor_ε 📖	mathematical	—	CategoryTheory.Functor.LaxMonoidal.ε CategoryTheory.Discrete CategoryTheory.discreteCategory addMonoidal addMonoidalFunctor CategoryTheory.Functor.Monoidal.toLaxMonoidal addMonoidalFunctorMonoidal eqToHom AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DFunLike.coe AddMonoidHom AddMonoidHom.instFunLike AddMonoidHom.map_zero	—	—
addMonoidalFunctor_η 📖	mathematical	—	CategoryTheory.Functor.OplaxMonoidal.η CategoryTheory.Discrete CategoryTheory.discreteCategory addMonoidal addMonoidalFunctor CategoryTheory.Functor.Monoidal.toOplaxMonoidal addMonoidalFunctorMonoidal eqToHom DFunLike.coe AddMonoidHom AddZeroClass.toAddZero AddMonoid.toAddZeroClass AddMonoidHom.instFunLike AddZero.toZero AddMonoidHom.map_zero	—	—
addMonoidalFunctor_μ 📖	mathematical	—	CategoryTheory.Functor.LaxMonoidal.μ CategoryTheory.Discrete CategoryTheory.discreteCategory addMonoidal addMonoidalFunctor CategoryTheory.Functor.Monoidal.toLaxMonoidal addMonoidalFunctorMonoidal eqToHom AddZero.toAdd AddZeroClass.toAddZero AddMonoid.toAddZeroClass DFunLike.coe AddMonoidHom AddMonoidHom.instFunLike as AddMonoidHom.map_add	—	—
addMonoidal_associator 📖	mathematical	—	CategoryTheory.MonoidalCategoryStruct.associator CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct addMonoidal eqToIso AddZero.toAdd AddZeroClass.toAddZero AddMonoid.toAddZeroClass as	—	—
addMonoidal_leftUnitor 📖	mathematical	—	CategoryTheory.MonoidalCategoryStruct.leftUnitor CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct addMonoidal eqToIso AddZero.toAdd AddZeroClass.toAddZero AddMonoid.toAddZeroClass as AddZero.toZero	—	—
addMonoidal_rightUnitor 📖	mathematical	—	CategoryTheory.MonoidalCategoryStruct.rightUnitor CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct addMonoidal eqToIso AddZero.toAdd AddZeroClass.toAddZero AddMonoid.toAddZeroClass as AddZero.toZero	—	—
addMonoidal_tensorObj_as 📖	mathematical	—	as CategoryTheory.MonoidalCategoryStruct.tensorObj CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct addMonoidal AddZero.toAdd AddZeroClass.toAddZero AddMonoid.toAddZeroClass	—	—
addMonoidal_tensorUnit_as 📖	mathematical	—	as CategoryTheory.MonoidalCategoryStruct.tensorUnit CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct addMonoidal AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass	—	—
monoidalFunctorComp_isMonoidal 📖	mathematical	—	CategoryTheory.NatTrans.IsMonoidal CategoryTheory.Discrete CategoryTheory.discreteCategory monoidal CategoryTheory.Functor.comp monoidalFunctor MonoidHom.comp MulOneClass.toMulOne Monoid.toMulOneClass CategoryTheory.Iso.hom CategoryTheory.Functor CategoryTheory.Functor.category monoidalFunctorComp CategoryTheory.Functor.LaxMonoidal.comp CategoryTheory.Functor.Monoidal.toLaxMonoidal monoidalFunctorMonoidal	—	MonoidHom.map_one CategoryTheory.eqToHom_map CategoryTheory.eqToHom_trans CategoryTheory.Category.comp_id MonoidHom.map_mul CategoryTheory.MonoidalCategory.tensorHom_id CategoryTheory.MonoidalCategory.id_whiskerRight CategoryTheory.Category.id_comp
monoidalFunctor_obj 📖	mathematical	—	CategoryTheory.Functor.obj CategoryTheory.Discrete CategoryTheory.discreteCategory monoidalFunctor DFunLike.coe MonoidHom MulOneClass.toMulOne Monoid.toMulOneClass MonoidHom.instFunLike	—	—
monoidalFunctor_δ 📖	mathematical	—	CategoryTheory.Functor.OplaxMonoidal.δ CategoryTheory.Discrete CategoryTheory.discreteCategory monoidal monoidalFunctor CategoryTheory.Functor.Monoidal.toOplaxMonoidal monoidalFunctorMonoidal eqToHom DFunLike.coe MonoidHom MulOneClass.toMulOne Monoid.toMulOneClass MonoidHom.instFunLike MulOne.toMul as MonoidHom.map_mul	—	—
monoidalFunctor_ε 📖	mathematical	—	CategoryTheory.Functor.LaxMonoidal.ε CategoryTheory.Discrete CategoryTheory.discreteCategory monoidal monoidalFunctor CategoryTheory.Functor.Monoidal.toLaxMonoidal monoidalFunctorMonoidal eqToHom MulOne.toOne MulOneClass.toMulOne Monoid.toMulOneClass DFunLike.coe MonoidHom MonoidHom.instFunLike MonoidHom.map_one	—	—
monoidalFunctor_η 📖	mathematical	—	CategoryTheory.Functor.OplaxMonoidal.η CategoryTheory.Discrete CategoryTheory.discreteCategory monoidal monoidalFunctor CategoryTheory.Functor.Monoidal.toOplaxMonoidal monoidalFunctorMonoidal eqToHom DFunLike.coe MonoidHom MulOneClass.toMulOne Monoid.toMulOneClass MonoidHom.instFunLike MulOne.toOne MonoidHom.map_one	—	—
monoidalFunctor_μ 📖	mathematical	—	CategoryTheory.Functor.LaxMonoidal.μ CategoryTheory.Discrete CategoryTheory.discreteCategory monoidal monoidalFunctor CategoryTheory.Functor.Monoidal.toLaxMonoidal monoidalFunctorMonoidal eqToHom MulOne.toMul MulOneClass.toMulOne Monoid.toMulOneClass DFunLike.coe MonoidHom MonoidHom.instFunLike as MonoidHom.map_mul	—	—
monoidal_associator 📖	mathematical	—	CategoryTheory.MonoidalCategoryStruct.associator CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct monoidal eqToIso MulOne.toMul MulOneClass.toMulOne Monoid.toMulOneClass as	—	—
monoidal_leftUnitor 📖	mathematical	—	CategoryTheory.MonoidalCategoryStruct.leftUnitor CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct monoidal eqToIso MulOne.toMul MulOneClass.toMulOne Monoid.toMulOneClass as MulOne.toOne	—	—
monoidal_rightUnitor 📖	mathematical	—	CategoryTheory.MonoidalCategoryStruct.rightUnitor CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct monoidal eqToIso MulOne.toMul MulOneClass.toMulOne Monoid.toMulOneClass as MulOne.toOne	—	—
monoidal_tensorObj_as 📖	mathematical	—	as CategoryTheory.MonoidalCategoryStruct.tensorObj CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct monoidal MulOne.toMul MulOneClass.toMulOne Monoid.toMulOneClass	—	—
monoidal_tensorUnit_as 📖	mathematical	—	as CategoryTheory.MonoidalCategoryStruct.tensorUnit CategoryTheory.Discrete CategoryTheory.discreteCategory CategoryTheory.MonoidalCategory.toMonoidalCategoryStruct monoidal MulOne.toOne MulOneClass.toMulOne Monoid.toMulOneClass	—	—

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