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Mathlib.CategoryTheory.Functor.Currying

Curry and uncurry, as functors. #

We define curry : ((C × D) ⥤ E) ⥤ (C ⥤ (D ⥤ E)) and uncurry : (C ⥤ (D ⥤ E)) ⥤ ((C × D) ⥤ E), and verify that they provide an equivalence of categories currying : (C ⥤ (D ⥤ E)) ≌ ((C × D) ⥤ E).

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def CategoryTheory.uncurry {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] :
CategoryTheory.Functor (CategoryTheory.Functor C (CategoryTheory.Functor D E)) (CategoryTheory.Functor (C × D) E)

The uncurrying functor, taking a functor C ⥤ (D ⥤ E) and producing a functor (C × D) ⥤ E.

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    theorem CategoryTheory.uncurry_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) {X Y : C × D} (f : X ⟶ Y) :
    (CategoryTheory.uncurry.obj F).map f = CategoryTheory.CategoryStruct.comp ((F.map f.1).app X.2) ((F.obj Y.1).map f.2)
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    theorem CategoryTheory.uncurry_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : C × D) :
    (CategoryTheory.uncurry.obj F).obj X = (F.obj X.1).obj X.2
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    theorem CategoryTheory.uncurry_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {X✝ Y✝ : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (T : X✝ ⟶ Y✝) (X : C × D) :
    (CategoryTheory.uncurry.map T).app X = (T.app X.1).app X.2
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    def CategoryTheory.curryObj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) :
    CategoryTheory.Functor C (CategoryTheory.Functor D E)

    The object level part of the currying functor. (See curry for the functorial version.)

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      def CategoryTheory.curry {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] :
      CategoryTheory.Functor (CategoryTheory.Functor (C × D) E) (CategoryTheory.Functor C (CategoryTheory.Functor D E))

      The currying functor, taking a functor (C × D) ⥤ E and producing a functor C ⥤ (D ⥤ E).

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        theorem CategoryTheory.curry_obj_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) {X✝ Y✝ : D} (g : X✝ ⟶ Y✝) :
        ((CategoryTheory.curry.obj F).obj X).map g = F.map (CategoryTheory.CategoryStruct.id X, g)
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        theorem CategoryTheory.curry_obj_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) (Y : D) :
        ((CategoryTheory.curry.obj F).obj X).obj Y = F.obj (X, Y)
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        theorem CategoryTheory.curry_obj_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) {X✝ Y✝ : C} (f : X✝ ⟶ Y✝) (Y : D) :
        ((CategoryTheory.curry.obj F).map f).app Y = F.map (f, CategoryTheory.CategoryStruct.id Y)
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        theorem CategoryTheory.curry_map_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {X✝ Y✝ : CategoryTheory.Functor (C × D) E} (T : X✝ ⟶ Y✝) (X : C) (Y : D) :
        ((CategoryTheory.curry.map T).app X).app Y = T.app (X, Y)
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        def CategoryTheory.currying {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] :
        CategoryTheory.Functor C (CategoryTheory.Functor D E) ≌ CategoryTheory.Functor (C × D) E

        The equivalence of functor categories given by currying/uncurrying.

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          theorem CategoryTheory.currying_inverse_obj_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) (Y : D) :
          ((CategoryTheory.currying.inverse.obj F).obj X).obj Y = F.obj (X, Y)
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          theorem CategoryTheory.currying_inverse_map_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {X✝ Y✝ : CategoryTheory.Functor (C × D) E} (T : X✝ ⟶ Y✝) (X : C) (Y : D) :
          ((CategoryTheory.currying.inverse.map T).app X).app Y = T.app (X, Y)
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          theorem CategoryTheory.currying_unitIso_inv_app_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : C) (X✝ : D) :
          ((CategoryTheory.currying.unitIso.inv.app X).app X✝).app X✝ = CategoryTheory.CategoryStruct.id ((X.obj X✝).obj X✝)
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          theorem CategoryTheory.currying_unitIso_hom_app_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : C) (X✝ : D) :
          ((CategoryTheory.currying.unitIso.hom.app X).app X✝).app X✝ = CategoryTheory.CategoryStruct.id ((X.obj X✝).obj X✝)
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          theorem CategoryTheory.currying_functor_obj_obj {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : C × D) :
          (CategoryTheory.currying.functor.obj F).obj X = (F.obj X.1).obj X.2
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          theorem CategoryTheory.currying_functor_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {X✝ Y✝ : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (T : X✝ ⟶ Y✝) (X : C × D) :
          (CategoryTheory.currying.functor.map T).app X = (T.app X.1).app X.2
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          theorem CategoryTheory.currying_inverse_obj_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) (X : C) {X✝ Y✝ : D} (g : X✝ ⟶ Y✝) :
          ((CategoryTheory.currying.inverse.obj F).obj X).map g = F.map (CategoryTheory.CategoryStruct.id X, g)
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          theorem CategoryTheory.currying_inverse_obj_map_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor (C × D) E) {X✝ Y✝ : C} (f : X✝ ⟶ Y✝) (Y : D) :
          ((CategoryTheory.currying.inverse.obj F).map f).app Y = F.map (f, CategoryTheory.CategoryStruct.id Y)
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          theorem CategoryTheory.currying_functor_obj_map {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) {X Y : C × D} (f : X ⟶ Y) :
          (CategoryTheory.currying.functor.obj F).map f = CategoryTheory.CategoryStruct.comp ((F.map f.1).app X.2) ((F.obj Y.1).map f.2)
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          theorem CategoryTheory.currying_counitIso_inv_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor (C × D) E) (X✝ : C × D) :
          (CategoryTheory.currying.counitIso.inv.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.obj (X✝.1, X✝.2))
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          theorem CategoryTheory.currying_counitIso_hom_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor (C × D) E) (X✝ : C × D) :
          (CategoryTheory.currying.counitIso.hom.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.obj (X✝.1, X✝.2))
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          def CategoryTheory.flipIsoCurrySwapUncurry {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) :
          F.flip ≅ CategoryTheory.curry.obj ((CategoryTheory.Prod.swap D C).comp (CategoryTheory.uncurry.obj F))

          F.flip is isomorphic to uncurrying F, swapping the variables, and currying.

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            theorem CategoryTheory.flipIsoCurrySwapUncurry_inv_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X✝ : C) :
            ((CategoryTheory.flipIsoCurrySwapUncurry F).inv.app X).app X✝ = CategoryTheory.CategoryStruct.id ((F.obj X✝).obj X)
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            theorem CategoryTheory.flipIsoCurrySwapUncurry_hom_app_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X✝ : C) :
            ((CategoryTheory.flipIsoCurrySwapUncurry F).hom.app X).app X✝ = CategoryTheory.CategoryStruct.id ((F.obj X✝).obj X)
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            def CategoryTheory.uncurryObjFlip {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) :
            CategoryTheory.uncurry.obj F.flip ≅ (CategoryTheory.Prod.swap D C).comp (CategoryTheory.uncurry.obj F)

            The uncurrying of F.flip is isomorphic to swapping the factors followed by the uncurrying of F.

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              theorem CategoryTheory.uncurryObjFlip_inv_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) :
              (CategoryTheory.uncurryObjFlip F).inv.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)
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              theorem CategoryTheory.uncurryObjFlip_hom_app {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) :
              (CategoryTheory.uncurryObjFlip F).hom.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)
              source
              def CategoryTheory.whiskeringRight₂ (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] :
              CategoryTheory.Functor (CategoryTheory.Functor C (CategoryTheory.Functor D E)) (CategoryTheory.Functor (CategoryTheory.Functor B C) (CategoryTheory.Functor (CategoryTheory.Functor B D) (CategoryTheory.Functor B E)))

              A version of CategoryTheory.whiskeringRight for bifunctors, obtained by uncurrying, applying whiskeringRight and currying back

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                theorem CategoryTheory.whiskeringRight₂_obj_map_app_app (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) {X✝ Y✝ : CategoryTheory.Functor B C} (f : X✝ ⟶ Y✝) (Y : CategoryTheory.Functor B D) (X✝¹ : B) :
                ((((CategoryTheory.whiskeringRight₂ B C D E).obj X).map f).app Y).app X✝¹ = (X.map (f.app X✝¹)).app (Y.obj X✝¹)
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                theorem CategoryTheory.whiskeringRight₂_obj_obj_map_app (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : CategoryTheory.Functor B C) {X✝¹ Y✝ : CategoryTheory.Functor B D} (g : X✝¹ ⟶ Y✝) (X✝ : B) :
                ((((CategoryTheory.whiskeringRight₂ B C D E).obj X).obj X✝).map g).app X✝ = (X.obj (X✝.obj X✝)).map (g.app X✝)
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                theorem CategoryTheory.whiskeringRight₂_obj_obj_obj_map (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : CategoryTheory.Functor B C) (Y : CategoryTheory.Functor B D) {X✝¹ Y✝ : B} (f : X✝¹ ⟶ Y✝) :
                ((((CategoryTheory.whiskeringRight₂ B C D E).obj X).obj X✝).obj Y).map f = CategoryTheory.CategoryStruct.comp ((X.map (X✝.map f)).app (Y.obj X✝¹)) ((X.obj (X✝.obj Y✝)).map (Y.map f))
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                theorem CategoryTheory.whiskeringRight₂_map_app_app_app (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] {X✝ Y✝ : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (f : X✝ ⟶ Y✝) (X : CategoryTheory.Functor B C) (Y : CategoryTheory.Functor B D) (c : B) :
                ((((CategoryTheory.whiskeringRight₂ B C D E).map f).app X).app Y).app c = (f.app (X.obj c)).app (Y.obj c)
                source
                @[simp]
                theorem CategoryTheory.whiskeringRight₂_obj_obj_obj_obj (B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : CategoryTheory.Functor B C) (Y : CategoryTheory.Functor B D) (X✝ : B) :
                ((((CategoryTheory.whiskeringRight₂ B C D E).obj X).obj X✝).obj Y).obj X✝ = (X.obj (X✝.obj X✝)).obj (Y.obj X✝)
                source
                theorem CategoryTheory.Functor.uncurry_obj_curry_obj {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor (B × C) D) :
                CategoryTheory.uncurry.obj (CategoryTheory.curry.obj F) = F
                source
                theorem CategoryTheory.Functor.curry_obj_injective {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {F₁ F₂ : CategoryTheory.Functor (C × D) E} (h : CategoryTheory.curry.obj F₁ = CategoryTheory.curry.obj F₂) :
                F₁ = F₂
                source
                theorem CategoryTheory.Functor.curry_obj_uncurry_obj {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor B (CategoryTheory.Functor C D)) :
                CategoryTheory.curry.obj (CategoryTheory.uncurry.obj F) = F
                source
                theorem CategoryTheory.Functor.uncurry_obj_injective {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {F₁ F₂ : CategoryTheory.Functor B (CategoryTheory.Functor C D)} (h : CategoryTheory.uncurry.obj F₁ = CategoryTheory.uncurry.obj F₂) :
                F₁ = F₂
                source
                theorem CategoryTheory.Functor.flip_flip {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor B (CategoryTheory.Functor C D)) :
                F.flip.flip = F
                source
                theorem CategoryTheory.Functor.flip_injective {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {F₁ F₂ : CategoryTheory.Functor B (CategoryTheory.Functor C D)} (h : F₁.flip = F₂.flip) :
                F₁ = F₂
                source
                theorem CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {H : Type u₅} [CategoryTheory.Category.{v₅, u₅} H] (F₁ : CategoryTheory.Functor B C) (F₂ : CategoryTheory.Functor D E) (G : CategoryTheory.Functor (C × E) H) :
                CategoryTheory.uncurry.obj (F₂.comp (F₁.comp (CategoryTheory.curry.obj G)).flip).flip = (F₁.prod F₂).comp G
                source
                theorem CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip' {B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {H : Type u₅} [CategoryTheory.Category.{v₅, u₅} H] (F₁ : CategoryTheory.Functor B C) (F₂ : CategoryTheory.Functor D E) (G : CategoryTheory.Functor (C × E) H) :
                CategoryTheory.uncurry.obj (F₁.comp (F₂.comp (CategoryTheory.curry.obj G).flip).flip) = (F₁.prod F₂).comp G