Documentation

Mathlib.CategoryTheory.Functor.Trifunctor

Trifunctors obtained by composition of bifunctors #

Given two bifunctors F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂ and G : C₁₂ ⥤ C₃ ⥤ C₄, we define the trifunctor bifunctorComp₁₂ F₁₂ G : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄ which sends three objects X₁ : C₁, X₂ : C₂ and X₃ : C₃ to G.obj ((F₁₂.obj X₁).obj X₂)).obj X₃.

Similarly, given two bifunctors F : C₁ ⥤ C₂₃ ⥤ C₄ and G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃, we define the trifunctor bifunctorComp₂₃ F G₂₃ : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄ which sends three objects X₁ : C₁, X₂ : C₂ and X₃ : C₃ to (F.obj X₁).obj ((G₂₃.obj X₂).obj X₃).

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def CategoryTheory.bifunctorComp₁₂Obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) (X₁ : C₁) :
Functor C₂ (Functor C₃ C₄)

Auxiliary definition for bifunctorComp₁₂.

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      theorem CategoryTheory.bifunctorComp₁₂Obj_map_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) (X₁ : C₁) {X₂ Y₂ : C₂} (φ : X₂ ⟶ Y₂) (X₃ : C₃) :
      ((bifunctorComp₁₂Obj F₁₂ G X₁).map φ).app X₃ = (G.map ((F₁₂.obj X₁).map φ)).app X₃
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      @[simp]
      theorem CategoryTheory.bifunctorComp₁₂Obj_obj_map {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) (X₁ : C₁) (X₂ : C₂) {x✝ x✝¹ : C₃} (φ : x✝ ⟶ x✝¹) :
      ((bifunctorComp₁₂Obj F₁₂ G X₁).obj X₂).map φ = (G.obj ((F₁₂.obj X₁).obj X₂)).map φ
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      @[simp]
      theorem CategoryTheory.bifunctorComp₁₂Obj_obj_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) (X₁ : C₁) (X₂ : C₂) (X₃ : C₃) :
      ((bifunctorComp₁₂Obj F₁₂ G X₁).obj X₂).obj X₃ = (G.obj ((F₁₂.obj X₁).obj X₂)).obj X₃
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      def CategoryTheory.bifunctorComp₁₂ {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) :
      Functor C₁ (Functor C₂ (Functor C₃ C₄))

      Given two bifunctors F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂ and G : C₁₂ ⥤ C₃ ⥤ C₄, this is the trifunctor C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄ obtained by composition.

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          theorem CategoryTheory.bifunctorComp₁₂_map_app_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) {X₁ Y₁ : C₁} (φ : X₁ ⟶ Y₁) (X₂ : C₂) (X₃ : C₃) :
          (((bifunctorComp₁₂ F₁₂ G).map φ).app X₂).app X₃ = (G.map ((F₁₂.map φ).app X₂)).app X₃
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          @[simp]
          theorem CategoryTheory.bifunctorComp₁₂_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) (X₁ : C₁) :
          (bifunctorComp₁₂ F₁₂ G).obj X₁ = bifunctorComp₁₂Obj F₁₂ G X₁
          source
          def CategoryTheory.bifunctorComp₁₂FunctorObj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) :
          Functor (Functor C₁₂ (Functor C₃ C₄)) (Functor C₁ (Functor C₂ (Functor C₃ C₄)))

          Auxiliary definition for bifunctorComp₁₂Functor.

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              theorem CategoryTheory.bifunctorComp₁₂FunctorObj_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) (G : Functor C₁₂ (Functor C₃ C₄)) :
              (bifunctorComp₁₂FunctorObj F₁₂).obj G = bifunctorComp₁₂ F₁₂ G
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              @[simp]
              theorem CategoryTheory.bifunctorComp₁₂FunctorObj_map_app_app_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) {G G' : Functor C₁₂ (Functor C₃ C₄)} (φ : G ⟶ G') (X₁ : C₁) (X₂ : C₂) (X₃ : C₃) :
              ((((bifunctorComp₁₂FunctorObj F₁₂).map φ).app X₁).app X₂).app X₃ = (φ.app ((F₁₂.obj X₁).obj X₂)).app X₃
              source
              def CategoryTheory.bifunctorComp₁₂FunctorMap {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] {F₁₂ F₁₂' : Functor C₁ (Functor C₂ C₁₂)} (φ : F₁₂ ⟶ F₁₂') :
              bifunctorComp₁₂FunctorObj F₁₂ ⟶ bifunctorComp₁₂FunctorObj F₁₂'

              Auxiliary definition for bifunctorComp₁₂Functor.

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                  theorem CategoryTheory.bifunctorComp₁₂FunctorMap_app_app_app_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] {F₁₂ F₁₂' : Functor C₁ (Functor C₂ C₁₂)} (φ : F₁₂ ⟶ F₁₂') (G : Functor C₁₂ (Functor C₃ C₄)) (X₁ : C₁) (X₂ : C₂) (X₃ : C₃) :
                  ((((bifunctorComp₁₂FunctorMap φ).app G).app X₁).app X₂).app X₃ = (G.map ((φ.app X₁).app X₂)).app X₃
                  source
                  def CategoryTheory.bifunctorComp₁₂Functor {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] :
                  Functor (Functor C₁ (Functor C₂ C₁₂)) (Functor (Functor C₁₂ (Functor C₃ C₄)) (Functor C₁ (Functor C₂ (Functor C₃ C₄))))

                  The functor (C₁ ⥤ C₂ ⥤ C₁₂) ⥤ (C₁₂ ⥤ C₃ ⥤ C₄) ⥤ C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄ which sends F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂ and G : C₁₂ ⥤ C₃ ⥤ C₄ to the functor bifunctorComp₁₂ F₁₂ G : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄.

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                      theorem CategoryTheory.bifunctorComp₁₂Functor_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] (F₁₂ : Functor C₁ (Functor C₂ C₁₂)) :
                      bifunctorComp₁₂Functor.obj F₁₂ = bifunctorComp₁₂FunctorObj F₁₂
                      source
                      @[simp]
                      theorem CategoryTheory.bifunctorComp₁₂Functor_map {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₁₂ : Type u_5} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_5} C₁₂] {X✝ Y✝ : Functor C₁ (Functor C₂ C₁₂)} (φ : X✝ ⟶ Y✝) :
                      bifunctorComp₁₂Functor.map φ = bifunctorComp₁₂FunctorMap φ
                      source
                      def CategoryTheory.bifunctorComp₂₃Obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) (X₁ : C₁) :
                      Functor C₂ (Functor C₃ C₄)

                      Auxiliary definition for bifunctorComp₂₃.

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                          theorem CategoryTheory.bifunctorComp₂₃Obj_obj_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) (X₁ : C₁) (X₂ : C₂) (X₃ : C₃) :
                          ((bifunctorComp₂₃Obj F G₂₃ X₁).obj X₂).obj X₃ = (F.obj X₁).obj ((G₂₃.obj X₂).obj X₃)
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                          @[simp]
                          theorem CategoryTheory.bifunctorComp₂₃Obj_obj_map {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) (X₁ : C₁) (X₂ : C₂) {X✝ Y✝ : C₃} (φ : X✝ ⟶ Y✝) :
                          ((bifunctorComp₂₃Obj F G₂₃ X₁).obj X₂).map φ = (F.obj X₁).map ((G₂₃.obj X₂).map φ)
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                          theorem CategoryTheory.bifunctorComp₂₃Obj_map_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) (X₁ : C₁) {X₂ Y₂ : C₂} (φ : X₂ ⟶ Y₂) (X₃ : C₃) :
                          ((bifunctorComp₂₃Obj F G₂₃ X₁).map φ).app X₃ = (F.obj X₁).map ((G₂₃.map φ).app X₃)
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                          def CategoryTheory.bifunctorComp₂₃ {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) :
                          Functor C₁ (Functor C₂ (Functor C₃ C₄))

                          Given two bifunctors F : C₁ ⥤ C₂₃ ⥤ C₄ and G₂₃ : C₂ ⥤ C₃ ⥤ C₄, this is the trifunctor C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄ obtained by composition.

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                              theorem CategoryTheory.bifunctorComp₂₃_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) (X₁ : C₁) :
                              (bifunctorComp₂₃ F G₂₃).obj X₁ = bifunctorComp₂₃Obj F G₂₃ X₁
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                              @[simp]
                              theorem CategoryTheory.bifunctorComp₂₃_map_app_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) {X₁ Y₁ : C₁} (φ : X₁ ⟶ Y₁) (X₂ : C₂) (X₃ : C₃) :
                              (((bifunctorComp₂₃ F G₂₃).map φ).app X₂).app X₃ = (F.map φ).app ((G₂₃.obj X₂).obj X₃)
                              source
                              def CategoryTheory.bifunctorComp₂₃FunctorObj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) :
                              Functor (Functor C₂ (Functor C₃ C₂₃)) (Functor C₁ (Functor C₂ (Functor C₃ C₄)))

                              Auxiliary definition for bifunctorComp₂₃Functor.

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                                  theorem CategoryTheory.bifunctorComp₂₃FunctorObj_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) :
                                  (bifunctorComp₂₃FunctorObj F).obj G₂₃ = bifunctorComp₂₃ F G₂₃
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                                  @[simp]
                                  theorem CategoryTheory.bifunctorComp₂₃FunctorObj_map_app_app_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) {G₂₃ G₂₃' : Functor C₂ (Functor C₃ C₂₃)} (φ : G₂₃ ⟶ G₂₃') (X₁ : C₁) (X₂ : C₂) (X₃ : C₃) :
                                  ((((bifunctorComp₂₃FunctorObj F).map φ).app X₁).app X₂).app X₃ = (F.obj X₁).map ((φ.app X₂).app X₃)
                                  source
                                  def CategoryTheory.bifunctorComp₂₃FunctorMap {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] {F F' : Functor C₁ (Functor C₂₃ C₄)} (φ : F ⟶ F') :
                                  bifunctorComp₂₃FunctorObj F ⟶ bifunctorComp₂₃FunctorObj F'

                                  Auxiliary definition for bifunctorComp₂₃Functor.

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                                      theorem CategoryTheory.bifunctorComp₂₃FunctorMap_app_app_app_app {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] {F F' : Functor C₁ (Functor C₂₃ C₄)} (φ : F ⟶ F') (G₂₃ : Functor C₂ (Functor C₃ C₂₃)) (X₁ : C₁) (X₂ : C₂) (X₃ : C₃) :
                                      ((((bifunctorComp₂₃FunctorMap φ).app G₂₃).app X₁).app X₂).app X₃ = (φ.app X₁).app ((G₂₃.obj X₂).obj X₃)
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                                      def CategoryTheory.bifunctorComp₂₃Functor {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] :
                                      Functor (Functor C₁ (Functor C₂₃ C₄)) (Functor (Functor C₂ (Functor C₃ C₂₃)) (Functor C₁ (Functor C₂ (Functor C₃ C₄))))

                                      The functor (C₁ ⥤ C₂₃ ⥤ C₄) ⥤ (C₂ ⥤ C₃ ⥤ C₂₃) ⥤ C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄ which sends F : C₁ ⥤ C₂₃ ⥤ C₄ and G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃ to the functor bifunctorComp₂₃ F G₂₃ : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄.

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                                          @[simp]
                                          theorem CategoryTheory.bifunctorComp₂₃Functor_obj {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] (F : Functor C₁ (Functor C₂₃ C₄)) :
                                          bifunctorComp₂₃Functor.obj F = bifunctorComp₂₃FunctorObj F
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                                          @[simp]
                                          theorem CategoryTheory.bifunctorComp₂₃Functor_map {C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_3} {C₄ : Type u_4} {C₂₃ : Type u_6} [Category.{u_7, u_1} C₁] [Category.{u_8, u_2} C₂] [Category.{u_9, u_3} C₃] [Category.{u_10, u_4} C₄] [Category.{u_11, u_6} C₂₃] {X✝ Y✝ : Functor C₁ (Functor C₂₃ C₄)} (φ : X✝ ⟶ Y✝) :
                                          bifunctorComp₂₃Functor.map φ = bifunctorComp₂₃FunctorMap φ