awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
f : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op X)
j j' : (Over.{v₁, u₁} (unop (op X)))ᵒᵖ
φ : j ⟶ j'
⊢ Eq (α := tensorObj.{v₃, u₃} (𝒞 := inst_1) (F.obj (op (unop j).left)) M ⟶ G.obj (op (unop j').left))
    ((F.map (Over.Hom.left φ.unop).op ⊗ₘ 𝟙 M) ≫
      uncurry (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j').hom.op)))
    (uncurry (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op)) ≫
      G.map (Over.Hom.left φ.unop).op)
have rw : Eq (α := tensorObj.{v₃, u₃} (𝒞 := inst_1) (F.obj (op (unop j).left)) M ⟶ G.obj (op (unop j').left))
  (uncurry
    (f ≫
      CategoryStruct.comp (Y := F.obj (op (unop j').left) ⟹ G.obj (op (unop j').left))
        (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j').hom.op))
        (eHomWhiskerRight A (F.map (Over.Hom.left φ.unop).op) (G.obj (op (unop j').left)))))
  (uncurry
    (f ≫
      CategoryStruct.comp (Y := F.obj (op (unop j).left) ⟹ G.obj (op (unop j).left))
        (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))
        (eHomWhiskerLeft A (F.obj (op (unop j).left)) (G.map (Over.Hom.left φ.unop).op))))
Tactic: rw [tensorHom_id, ← uncurry_natural_right, ← uncurry_pre_app, Category.assoc,
          Category.assoc, ← enrichedOrdinaryCategorySelf_eHomWhiskerRight,
          ← enrichedOrdinaryCategorySelf_eHomWhiskerLeft]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
f : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op X)
j j' : (Over.{v₁, u₁} (unop (op X)))ᵒᵖ
φ : j ⟶ j'
⊢ Eq (α := tensorObj.{v₃, u₃} (𝒞 := inst_1) (F.obj (op (unop j).left)) M ⟶ G.obj (op (unop j').left))
    (uncurry
      (f ≫
        CategoryStruct.comp (Y := F.obj (op (unop j').left) ⟹ G.obj (op (unop j').left))
          (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j').hom.op))
          (eHomWhiskerRight A (F.map (Over.Hom.left φ.unop).op) (G.obj (op (unop j').left)))))
    (uncurry
      (f ≫
        CategoryStruct.comp (Y := F.obj (op (unop j).left) ⟹ G.obj (op (unop j).left))
          (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))
          (eHomWhiskerLeft A (F.obj (op (unop j).left)) (G.map (Over.Hom.left φ.unop).op))))
have e_a : Eq (α :=
  enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) ⟶
    F.obj (op (unop j).left) ⟹ G.obj (op (unop j').left))
  (CategoryStruct.comp (Y := F.obj (op (unop j').left) ⟹ G.obj (op (unop j').left))
    (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j').hom.op))
    (eHomWhiskerRight A (F.map (Over.Hom.left φ.unop).op) (G.obj (op (unop j').left))))
  (CategoryStruct.comp (Y := F.obj (op (unop j).left) ⟹ G.obj (op (unop j).left))
    (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))
    (eHomWhiskerLeft A (F.obj (op (unop j).left)) (G.map (Over.Hom.left φ.unop).op)))
Tactic: congr 2
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)
j j' : Under.{v₁, u₁} (op X)
φ : j ⟶ j'
⊢ Eq (α :=
    unop (op M) ⟶ Functor.obj (Functor.obj (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) (op j)) j')
    ((fun (j : Under.{v₁, u₁} (op X)) ↦
          (curry (g.app (op (Over.mk j.hom.unop))) :
            unop (op M) ⟶ unop ((Under.forget (op X) ⋙ F).op.obj (op j)) ⟹ (Under.forget (op X) ⋙ G).obj j))
        j ≫
      Functor.map (Functor.obj (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) (op j)) φ)
    ((fun (j : Under.{v₁, u₁} (op X)) ↦
          (curry (g.app (op (Over.mk j.hom.unop))) :
            unop (op M) ⟶ unop ((Under.forget (op X) ⋙ F).op.obj (op j)) ⟹ (Under.forget (op X) ⋙ G).obj j))
        j' ≫
      NatTrans.app (Functor.map (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) φ.op) j')
have dsimp : Eq (α := M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
  (CategoryStruct.comp (Y := @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j.right))
      (curry (g.app (op (Over.mk j.hom.unop))) : M ⟶ F.obj j.right ⟹ G.obj j.right)
      (eHomWhiskerLeft A (F.obj j.right) (G.map (Under.Hom.right φ)) :
        Quiver.Hom (V := A) (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j.right))
          (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))) :
    M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
  (CategoryStruct.comp (Y := @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j'.right) (G.obj j'.right))
      (curry (g.app (op (Over.mk j'.hom.unop))) : M ⟶ F.obj j'.right ⟹ G.obj j'.right)
      (eHomWhiskerRight A (F.map (Under.Hom.right φ)) (G.obj j'.right) :
        Quiver.Hom (V := A) (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j'.right) (G.obj j'.right))
          (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))) :
    M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)
j j' : Under.{v₁, u₁} (op X)
φ : j ⟶ j'
⊢ Eq (α := M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
    (CategoryStruct.comp (Y := @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j.right))
        (curry (g.app (op (Over.mk j.hom.unop))) : M ⟶ F.obj j.right ⟹ G.obj j.right)
        (eHomWhiskerLeft A (F.obj j.right) (G.map (Under.Hom.right φ)) :
          Quiver.Hom (V := A) (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j.right))
            (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))) :
      M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
    (CategoryStruct.comp (Y := @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j'.right) (G.obj j'.right))
        (curry (g.app (op (Over.mk j'.hom.unop))) : M ⟶ F.obj j'.right ⟹ G.obj j'.right)
        (eHomWhiskerRight A (F.map (Under.Hom.right φ)) (G.obj j'.right) :
          Quiver.Hom (V := A) (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j'.right) (G.obj j'.right))
            (@EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))) :
      M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
have rw : Eq (α := M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
  (curry (g.app (op (Over.mk j.hom.unop)) ≫ G.map (Under.Hom.right φ)) : M ⟶ F.obj j.right ⟹ G.obj j'.right)
  (curry (F.map (Under.Hom.right φ) ▷ M ≫ g.app (op (Over.mk j'.hom.unop))) : M ⟶ F.obj j.right ⟹ G.obj j'.right)
Tactic: rw [enrichedOrdinaryCategorySelf_eHomWhiskerRight,
        enrichedOrdinaryCategorySelf_eHomWhiskerLeft,
        curry_pre_app, ← curry_natural_right]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)
j j' : Under.{v₁, u₁} (op X)
φ : j ⟶ j'
⊢ Eq (α := M ⟶ @EnrichedCategory.Hom.{v₃, u₃, u₃} _ inst_1 inst_2 _ _ (F.obj j.right) (G.obj j'.right))
    (curry (g.app (op (Over.mk j.hom.unop)) ≫ G.map (Under.Hom.right φ)) : M ⟶ F.obj j.right ⟹ G.obj j'.right)
    (curry (F.map (Under.Hom.right φ) ▷ M ≫ g.app (op (Over.mk j'.hom.unop))) : M ⟶ F.obj j.right ⟹ G.obj j'.right)
have e_a : Eq (α := tensorObj.{v₃, u₃} (𝒞 := inst_1) (F.obj j.right) M ⟶ G.obj j'.right)
  (g.app (op (Over.mk j.hom.unop)) ≫ G.map (Under.Hom.right φ) :
    tensorObj.{v₃, u₃} (𝒞 := inst_1) (F.obj j.right) M ⟶ G.obj j'.right)
  (F.map (Under.Hom.right φ) ▷ M ≫ g.app (op (Over.mk j'.hom.unop)) :
    tensorObj.{v₃, u₃} (𝒞 := inst_1) (F.obj j.right) M ⟶ G.obj j'.right)
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
f : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op X)
⊢ (fun (g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)) ↦
        (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) ⋯ :
          unop (op M) ⟶ end_ (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))))
      ((fun (f : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op X)) ↦
          ({
              app := fun (j : (Over.{v₁, u₁} (unop (op X)))ᵒᵖ) ↦
                uncurry
                  (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op)),
              naturality := ⋯ } :
            NatTrans
              (Functor.op (Over.forget (unop (op X))) ⋙
                tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M))
              (Functor.op (Over.forget (unop (op X))) ⋙ G)))
        f) =
    f
have dsimp : end_.lift
    (fun (j : Under.{v₁, u₁} (op X)) ↦
      curry (uncurry (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk j.hom))))
    sorry =
  f
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
f : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op X)
⊢ end_.lift
      (fun (j : Under.{v₁, u₁} (op X)) ↦
        curry (uncurry (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk j.hom))))
      ⋯ =
    f
have h : ∀ (j : Under.{v₁, u₁} (op X)),
  CategoryStruct.comp (Y := end_ (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)))
      (end_.lift
        (fun (j : Under.{v₁, u₁} (op X)) ↦
          curry (uncurry (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk j.hom))))
        sorry)
      (end_.π (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) j) =
    f ≫ end_.π (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) j
Tactic: ext j
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
f : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op X)
j : Under.{v₁, u₁} (op X)
⊢ CategoryStruct.comp (Y := end_ (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)))
      (end_.lift
        (fun (j : Under.{v₁, u₁} (op X)) ↦
          curry (uncurry (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk j.hom))))
        ⋯)
      (end_.π (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) j) =
    f ≫ end_.π (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) j
have h : f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk j.hom) =
  f ≫ end_.π (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) j
Tactic: simp only [curry_uncurry, end_.lift_π]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)
⊢ (fun (f : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op X)) ↦
        ({
            app := fun (j : (Over.{v₁, u₁} (unop (op X)))ᵒᵖ) ↦
              uncurry
                (f ≫ enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op)),
            naturality := ⋯ } :
          NatTrans
            (Functor.op (Over.forget (unop (op X))) ⋙
              tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M))
            (Functor.op (Over.forget (unop (op X))) ⋙ G)))
      ((fun (g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)) ↦
          (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) ⋯ :
            unop (op M) ⟶ end_ (diagram A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))))
        g) =
    g
have dsimp : {
    app := fun (j : (Over.{v₁, u₁} X)ᵒᵖ) ↦
      uncurry
        (CategoryStruct.comp (Y := enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))
          (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) sorry)
          (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))),
    naturality := sorry } =
  g
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)
⊢ {
      app := fun (j : (Over.{v₁, u₁} X)ᵒᵖ) ↦
        uncurry
          (CategoryStruct.comp (Y := enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))
            (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) ⋯)
            (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))),
      naturality := ⋯ } =
    g
have w : ∀ (j : (Over.{v₁, u₁} X)ᵒᵖ),
  NatTrans.app
      {
        app := fun (j : (Over.{v₁, u₁} X)ᵒᵖ) ↦
          uncurry
            (CategoryStruct.comp (Y := enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))
              (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) sorry)
              (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))),
        naturality := sorry }
      j =
    g.app j
Tactic: ext j
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)
j : (Over.{v₁, u₁} X)ᵒᵖ
⊢ NatTrans.app
      {
        app := fun (j : (Over.{v₁, u₁} X)ᵒᵖ) ↦
          uncurry
            (CategoryStruct.comp (Y := enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))
              (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) ⋯)
              (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))),
        naturality := ⋯ }
      j =
    g.app j
have w : uncurry
    (CategoryStruct.comp (Y := enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))
      (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) sorry)
      (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))) =
  g.app j
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
J : GrothendieckTopology.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
inst_4 : HasFunctorEnrichedHom A F G
X : C
g : (presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).obj (op X)
j : (Over.{v₁, u₁} X)ᵒᵖ
⊢ uncurry
      (CategoryStruct.comp (Y := enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G))
        (end_.lift (fun (j : Under.{v₁, u₁} (op X)) ↦ curry (g.app (op (Over.mk j.hom.unop)))) ⋯)
        (enrichedHomπ A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G) (Under.mk (unop j).hom.op))) =
    g.app j
have w : g.app
    (op
      (Over.mk (Y := unop (Under.right.{v₁, u₁} (X := op X) (Under.mk (unop j).hom.op)))
        (Quiver.Hom.unop (Under.hom (Under.mk (unop j).hom.op))))) =
  g.app j
Tactic: simp only [uncurry_curry, end_.lift_π]
-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Monoidal (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
A : Type u₃
inst_1 : Category.{v₃, u₃} A
inst_2 : MonoidalCategory.{v₃, u₃} A
inst_3 : MonoidalClosed.{v₃, u₃} A
M : A
F G : Functor.{v₁, v₃, u₁, u₃} Cᵒᵖ A
X Y : C
f : X ⟶ Y
inst_4 : HasFunctorEnrichedHom A F G
y : (functorEnrichedHom A F G ⋙ Functor.obj coyoneda (op M)).obj (op Y)
⊢ Eq (α :=
    Functor.op (Over.forget X) ⋙ tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M) ⟶
      Functor.op (Over.forget X) ⋙ G)
    ((functorEnrichedHomCoyonedaObjEquiv M F G X :
          (a : M ⟶ enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) →
            (Over.forget X).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget X).op ⋙ G)
        (y ≫
          precompEnrichedHom' A (Under.map f.op) (Iso.refl (Under.map f.op ⋙ Under.forget (op Y) ⋙ F))
            (Iso.refl (Under.map f.op ⋙ Under.forget (op Y) ⋙ G))) :
      Functor.op (Over.forget X) ⋙ tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M) ⟶
        Functor.op (Over.forget X) ⋙ G)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            ((presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).map f.op) :
          (a : (Over.forget Y).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget Y).op ⋙ G) →
            (Over.forget X).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget X).op ⋙ G)
        ((functorEnrichedHomCoyonedaObjEquiv M F G Y :
            (a : M ⟶ enrichedHom A (Under.forget (op Y) ⋙ F) (Under.forget (op Y) ⋙ G)) →
              (Over.forget Y).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget Y).op ⋙ G)
          y) :
      Functor.op (Over.forget X) ⋙ tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M) ⟶
        Functor.op (Over.forget X) ⋙ G)
have w : ∀ (j : Over.{v₁, u₁} X),
  Eq.{v₃ + 1} (α :=
    (Functor.op (Over.forget X) ⋙ tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)).obj (op j) ⟶
      (Functor.op (Over.forget X) ⋙ G).obj (op j))
    (NatTrans.app
      ((functorEnrichedHomCoyonedaObjEquiv M F G X :
          (a : M ⟶ enrichedHom A (Under.forget (op X) ⋙ F) (Under.forget (op X) ⋙ G)) →
            (Over.forget X).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget X).op ⋙ G)
        (y ≫
          precompEnrichedHom' A (Under.map f.op) (Iso.refl (Under.map f.op ⋙ Under.forget (op Y) ⋙ F))
            (Iso.refl (Under.map f.op ⋙ Under.forget (op Y) ⋙ G))))
      (op j))
    (NatTrans.app
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
            ((presheafHom (tensorObj (𝒞 := Functor.category) F (Functor.obj (Functor.const Cᵒᵖ) M)) G).map f.op) :
          (a : (Over.forget Y).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget Y).op ⋙ G) →
            (Over.forget X).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget X).op ⋙ G)
        ((functorEnrichedHomCoyonedaObjEquiv M F G Y :
            (a : M ⟶ enrichedHom A (Under.forget (op Y) ⋙ F) (Under.forget (op Y) ⋙ G)) →
              (Over.forget Y).op ⋙ (F ⊗ (Functor.const Cᵒᵖ).obj M) ⟶ (Over.forget Y).op ⋙ G)
          y))
      (op j))
Tactic: ext ⟨j⟩