Documentation

Mathlib.CategoryTheory.Limits.Preserves.Limits

Isomorphisms about functors which preserve (co)limits #

If G preserves limits, and C and D have limits, then for any diagram F : J ⥤ C we have a canonical isomorphism preservesLimitsIso : G.obj (Limit F) ≅ Limit (F ⋙ G). We also show that we can commute IsLimit.lift of a preserved limit with Functor.mapCone: (PreservesLimit.preserves t).lift (G.mapCone c₂) = G.map (t.lift c₂).

The duals of these are also given. For functors which preserve (co)limits of specific shapes, see preserves/shapes.lean.

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@[simp]
theorem CategoryTheory.preserves_lift_mapCone {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] (c₁ c₂ : CategoryTheory.Limits.Cone F) (t : CategoryTheory.Limits.IsLimit c₁) :
(CategoryTheory.Limits.isLimitOfPreserves G t).lift (G.mapCone c₂) = G.map (t.lift c₂)
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def CategoryTheory.preservesLimitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] :
G.obj (CategoryTheory.Limits.limit F) ≅ CategoryTheory.Limits.limit (F.comp G)

If G preserves limits, we have an isomorphism from the image of the limit of a functor F to the limit of the functor F ⋙ G.

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    @[simp]
    theorem CategoryTheory.preservesLimitIso_hom_π {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (j : J) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso G F).hom (CategoryTheory.Limits.limit.π (F.comp G) j) = G.map (CategoryTheory.Limits.limit.π F j)
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    @[simp]
    theorem CategoryTheory.preservesLimitIso_hom_π_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (j : J) {Z : D} (h : G.obj (F.obj j) ⟶ Z) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso G F).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.comp G) j) h) = CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.limit.π F j)) h
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    @[deprecated CategoryTheory.preservesLimitIso_hom_π]
    theorem CategoryTheory.preservesLimitsIso_hom_π {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (j : J) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso G F).hom (CategoryTheory.Limits.limit.π (F.comp G) j) = G.map (CategoryTheory.Limits.limit.π F j)

    Alias of CategoryTheory.preservesLimitIso_hom_π.

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    @[simp]
    theorem CategoryTheory.preservesLimitIso_inv_π {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (j : J) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso G F).inv (G.map (CategoryTheory.Limits.limit.π F j)) = CategoryTheory.Limits.limit.π (F.comp G) j
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    @[simp]
    theorem CategoryTheory.preservesLimitIso_inv_π_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (j : J) {Z : D} (h : G.obj (F.obj j) ⟶ Z) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso G F).inv (CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.limit.π F j)) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.comp G) j) h
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    @[deprecated CategoryTheory.preservesLimitIso_inv_π]
    theorem CategoryTheory.preservesLimitsIso_inv_π {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (j : J) :
    CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso G F).inv (G.map (CategoryTheory.Limits.limit.π F j)) = CategoryTheory.Limits.limit.π (F.comp G) j

    Alias of CategoryTheory.preservesLimitIso_inv_π.

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    @[simp]
    theorem CategoryTheory.lift_comp_preservesLimitIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (t : CategoryTheory.Limits.Cone F) :
    CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.limit.lift F t)) (CategoryTheory.preservesLimitIso G F).hom = CategoryTheory.Limits.limit.lift (F.comp G) (G.mapCone t)
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    @[simp]
    theorem CategoryTheory.lift_comp_preservesLimitIso_hom_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (t : CategoryTheory.Limits.Cone F) {Z : D} (h : CategoryTheory.Limits.limit (F.comp G) ⟶ Z) :
    CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.limit.lift F t)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesLimitIso G F).hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.lift (F.comp G) (G.mapCone t)) h
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    @[deprecated CategoryTheory.lift_comp_preservesLimitIso_hom]
    theorem CategoryTheory.lift_comp_preservesLimitsIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] (t : CategoryTheory.Limits.Cone F) :
    CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.limit.lift F t)) (CategoryTheory.preservesLimitIso G F).hom = CategoryTheory.Limits.limit.lift (F.comp G) (G.mapCone t)

    Alias of CategoryTheory.lift_comp_preservesLimitIso_hom.

    source
    instance CategoryTheory.instIsIsoPost {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesLimit F G] [CategoryTheory.Limits.HasLimit F] :
    CategoryTheory.IsIso (CategoryTheory.Limits.limit.post F G)
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    def CategoryTheory.preservesLimitNatIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] [CategoryTheory.Limits.PreservesLimitsOfShape J G] [CategoryTheory.Limits.HasLimitsOfShape J D] [CategoryTheory.Limits.HasLimitsOfShape J C] :
    CategoryTheory.Limits.lim.comp G ≅ ((CategoryTheory.whiskeringRight J C D).obj G).comp CategoryTheory.Limits.lim

    If C, D has all limits of shape J, and G preserves them, then preservesLimitsIso is functorial wrt F.

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      @[simp]
      theorem CategoryTheory.preservesLimitNatIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] [CategoryTheory.Limits.PreservesLimitsOfShape J G] [CategoryTheory.Limits.HasLimitsOfShape J D] [CategoryTheory.Limits.HasLimitsOfShape J C] (X : CategoryTheory.Functor J C) :
      (CategoryTheory.preservesLimitNatIso G).inv.app X = (CategoryTheory.preservesLimitIso G X).inv
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      @[simp]
      theorem CategoryTheory.preservesLimitNatIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] [CategoryTheory.Limits.PreservesLimitsOfShape J G] [CategoryTheory.Limits.HasLimitsOfShape J D] [CategoryTheory.Limits.HasLimitsOfShape J C] (X : CategoryTheory.Functor J C) :
      (CategoryTheory.preservesLimitNatIso G).hom.app X = (CategoryTheory.preservesLimitIso G X).hom
      source
      theorem CategoryTheory.preservesLimit_of_isIso_post {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.HasLimit F] [CategoryTheory.Limits.HasLimit (F.comp G)] [CategoryTheory.IsIso (CategoryTheory.Limits.limit.post F G)] :
      CategoryTheory.Limits.PreservesLimit F G

      If the comparison morphism G.obj (limit F) ⟶ limit (F ⋙ G) is an isomorphism, then G preserves limits of F.

      source
      @[simp]
      theorem CategoryTheory.preserves_desc_mapCocone {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] (c₁ c₂ : CategoryTheory.Limits.Cocone F) (t : CategoryTheory.Limits.IsColimit c₁) :
      (CategoryTheory.Limits.isColimitOfPreserves G t).desc (G.mapCocone c₂) = G.map (t.desc c₂)
      source
      def CategoryTheory.preservesColimitIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] :
      G.obj (CategoryTheory.Limits.colimit F) ≅ CategoryTheory.Limits.colimit (F.comp G)

      If G preserves colimits, we have an isomorphism from the image of the colimit of a functor F to the colimit of the functor F ⋙ G.

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        @[simp]
        theorem CategoryTheory.ι_preservesColimitIso_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (j : J) :
        CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.comp G) j) (CategoryTheory.preservesColimitIso G F).inv = G.map (CategoryTheory.Limits.colimit.ι F j)
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        @[simp]
        theorem CategoryTheory.ι_preservesColimitIso_inv_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (j : J) {Z : D} (h : G.obj (CategoryTheory.Limits.colimit F) ⟶ Z) :
        CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.comp G) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesColimitIso G F).inv h) = CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.colimit.ι F j)) h
        source
        @[deprecated CategoryTheory.ι_preservesColimitIso_inv]
        theorem CategoryTheory.ι_preservesColimitsIso_inv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (j : J) :
        CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.comp G) j) (CategoryTheory.preservesColimitIso G F).inv = G.map (CategoryTheory.Limits.colimit.ι F j)

        Alias of CategoryTheory.ι_preservesColimitIso_inv.

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        @[simp]
        theorem CategoryTheory.ι_preservesColimitIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (j : J) :
        CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.colimit.ι F j)) (CategoryTheory.preservesColimitIso G F).hom = CategoryTheory.Limits.colimit.ι (F.comp G) j
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        @[simp]
        theorem CategoryTheory.ι_preservesColimitIso_hom_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (j : J) {Z : D} (h : CategoryTheory.Limits.colimit (F.comp G) ⟶ Z) :
        CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.colimit.ι F j)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesColimitIso G F).hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.comp G) j) h
        source
        @[deprecated CategoryTheory.ι_preservesColimitIso_hom]
        theorem CategoryTheory.ι_preservesColimitsIso_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (j : J) :
        CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.colimit.ι F j)) (CategoryTheory.preservesColimitIso G F).hom = CategoryTheory.Limits.colimit.ι (F.comp G) j

        Alias of CategoryTheory.ι_preservesColimitIso_hom.

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        @[simp]
        theorem CategoryTheory.preservesColimitIso_inv_comp_desc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (t : CategoryTheory.Limits.Cocone F) :
        CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesColimitIso G F).inv (G.map (CategoryTheory.Limits.colimit.desc F t)) = CategoryTheory.Limits.colimit.desc (F.comp G) (G.mapCocone t)
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        @[simp]
        theorem CategoryTheory.preservesColimitIso_inv_comp_desc_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (t : CategoryTheory.Limits.Cocone F) {Z : D} (h : G.obj t.pt ⟶ Z) :
        CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesColimitIso G F).inv (CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.colimit.desc F t)) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.desc (F.comp G) (G.mapCocone t)) h
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        @[deprecated CategoryTheory.preservesColimitIso_inv_comp_desc]
        theorem CategoryTheory.preservesColimitsIso_inv_comp_desc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] (t : CategoryTheory.Limits.Cocone F) :
        CategoryTheory.CategoryStruct.comp (CategoryTheory.preservesColimitIso G F).inv (G.map (CategoryTheory.Limits.colimit.desc F t)) = CategoryTheory.Limits.colimit.desc (F.comp G) (G.mapCocone t)

        Alias of CategoryTheory.preservesColimitIso_inv_comp_desc.

        source
        instance CategoryTheory.instIsIsoPost_1 {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.PreservesColimit F G] [CategoryTheory.Limits.HasColimit F] :
        CategoryTheory.IsIso (CategoryTheory.Limits.colimit.post F G)
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        • ⋯ = ⋯
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        def CategoryTheory.preservesColimitNatIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] [CategoryTheory.Limits.PreservesColimitsOfShape J G] [CategoryTheory.Limits.HasColimitsOfShape J D] [CategoryTheory.Limits.HasColimitsOfShape J C] :
        CategoryTheory.Limits.colim.comp G ≅ ((CategoryTheory.whiskeringRight J C D).obj G).comp CategoryTheory.Limits.colim

        If C, D has all colimits of shape J, and G preserves them, then preservesColimitIso is functorial wrt F.

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          @[simp]
          theorem CategoryTheory.preservesColimitNatIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] [CategoryTheory.Limits.PreservesColimitsOfShape J G] [CategoryTheory.Limits.HasColimitsOfShape J D] [CategoryTheory.Limits.HasColimitsOfShape J C] (X : CategoryTheory.Functor J C) :
          (CategoryTheory.preservesColimitNatIso G).hom.app X = (CategoryTheory.preservesColimitIso G X).hom
          source
          @[simp]
          theorem CategoryTheory.preservesColimitNatIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] [CategoryTheory.Limits.PreservesColimitsOfShape J G] [CategoryTheory.Limits.HasColimitsOfShape J D] [CategoryTheory.Limits.HasColimitsOfShape J C] (X : CategoryTheory.Functor J C) :
          (CategoryTheory.preservesColimitNatIso G).inv.app X = (CategoryTheory.preservesColimitIso G X).inv
          source
          theorem CategoryTheory.preservesColimit_of_isIso_post {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) {J : Type w} [CategoryTheory.Category.{w', w} J] (F : CategoryTheory.Functor J C) [CategoryTheory.Limits.HasColimit F] [CategoryTheory.Limits.HasColimit (F.comp G)] [CategoryTheory.IsIso (CategoryTheory.Limits.colimit.post F G)] :
          CategoryTheory.Limits.PreservesColimit F G

          If the comparison morphism colimit (F ⋙ G) ⟶ G.obj (colimit F) is an isomorphism, then G preserves colimits of F.