awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
X Y : Functor.{v, v', u, u'} Cᵒᵖ A
x : X ⟶ Y
x_1 : Functor.{v, v', u, u'} Cᵒᵖ A
X_1 : C
x_2 : Φ.fiber.obj X_1
⊢ Eq (α :=
    (tensorObj (𝒞 := category) X x_1).obj (Opposite.op X_1) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber Y) (obj Φ.presheafFiber x_1))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) X x_1))
      (Φ.toPresheafFiber X_1 x_2 (tensorObj (𝒞 := category) X x_1))
      (Φ.presheafFiberDesc
          (fun (X_2 : C) (x : Φ.fiber.obj X_2) ↦ Φ.toPresheafFiber X_2 x X ⊗ₘ Φ.toPresheafFiber X_2 x x_1) ⋯ ≫
        map Φ.presheafFiber x ▷ obj Φ.presheafFiber x_1))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) X x_1))
      (Φ.toPresheafFiber X_1 x_2 (tensorObj (𝒞 := category) X x_1))
      (map Φ.presheafFiber (x ▷ x_1) ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x Y ⊗ₘ Φ.toPresheafFiber X x x_1) ⋯))

Tactic:
simp [tensorHom_def', ← comp_whiskerRight, ← whisker_exchange_assoc]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.MonoidalCategory.whisker_exchange_assoc, CategoryTheory.MonoidalCategory.tensorHom_def'
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
X Y P : Functor.{v, v', u, u'} Cᵒᵖ A
f : X ⟶ Y
h :
  ∀ (X_1 : C) (x : Φ.fiber.obj X_1),
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) P X))
        (Φ.toPresheafFiber X_1 x (tensorObj (𝒞 := category) P X))
        (Φ.presheafFiberDesc
            (fun (X_2 : C) (x : Φ.fiber.obj X_2) ↦ Φ.toPresheafFiber X_2 x P ⊗ₘ Φ.toPresheafFiber X_2 x X) ⋯ ≫
          obj Φ.presheafFiber P ◁ map Φ.presheafFiber f) =
      CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) P X))
        (Φ.toPresheafFiber X_1 x (tensorObj (𝒞 := category) P X))
        (map Φ.presheafFiber (P ◁ f) ≫
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P ⊗ₘ Φ.toPresheafFiber X x Y) ⋯)
⊢ Eq (α :=
    obj Φ.presheafFiber (tensorObj (𝒞 := category) P X) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P) (obj Φ.presheafFiber Y))
    (Φ.presheafFiberDesc (fun (X_1 : C) (x : Φ.fiber.obj X_1) ↦ Φ.toPresheafFiber X_1 x P ⊗ₘ Φ.toPresheafFiber X_1 x X)
        ⋯ ≫
      obj Φ.presheafFiber P ◁ map Φ.presheafFiber f)
    (map Φ.presheafFiber (P ◁ f) ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P ⊗ₘ Φ.toPresheafFiber X x Y) ⋯)

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
X Y P : Functor.{v, v', u, u'} Cᵒᵖ A
f : X ⟶ Y
X_1 : C
x : Φ.fiber.obj X_1
⊢ CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) P X))
      (Φ.toPresheafFiber X_1 x (tensorObj (𝒞 := category) P X))
      (Φ.presheafFiberDesc
          (fun (X_2 : C) (x : Φ.fiber.obj X_2) ↦ Φ.toPresheafFiber X_2 x P ⊗ₘ Φ.toPresheafFiber X_2 x X) ⋯ ≫
        obj Φ.presheafFiber P ◁ map Φ.presheafFiber f) =
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) P X))
      (Φ.toPresheafFiber X_1 x (tensorObj (𝒞 := category) P X))
      (map Φ.presheafFiber (P ◁ f) ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P ⊗ₘ Φ.toPresheafFiber X x Y) ⋯)

Tactic:
simp [tensorHom_def, ← MonoidalCategory.whiskerLeft_comp, whisker_exchange_assoc]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonoidalCategory.tensorHom_def, CategoryTheory.MonoidalCategory.whisker_exchange_assoc
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
h :
  ∀ (X : C) (x : Φ.fiber.obj X),
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
        (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
        (Φ.presheafFiberDesc
            (fun (X : C) (x : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
            ⋯ ≫
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂)
                ⋯ ▷
              obj Φ.presheafFiber P₃ ≫
            Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))) =
      CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
        (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
        (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
          Φ.presheafFiberDesc
              (fun (X : C) (x : Φ.fiber.obj X) ↦
                Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
              ⋯ ≫
            obj Φ.presheafFiber P₁ ◁
              Φ.presheafFiberDesc
                (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃) ⋯)
⊢ Eq (α :=
    obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₁)
        (tensorObj.{v', u'} (𝒞 := inst_1) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
    (Φ.presheafFiberDesc
        (fun (X : C) (x : Φ.fiber.obj X) ↦
          Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
        ⋯ ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
          obj Φ.presheafFiber P₃ ≫
        Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          ⋯ ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            ⋯)

Tactic:
ext X x
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
h :
  CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
          ⋯ ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
            obj Φ.presheafFiber P₃ ≫
          Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))) =
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
            ⋯ ≫
          obj Φ.presheafFiber P₁ ◁
            Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
              ⋯)
⊢ CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
          ⋯ ≫
        Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
            obj Φ.presheafFiber P₃ ≫
          Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))) =
    CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
      (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
            ⋯ ≫
          obj Φ.presheafFiber P₁ ◁
            Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
              ⋯)

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
simp :
  tensorObj.{v', u'} (𝒞 := inst_1) (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) ◁ Φ.toPresheafFiber X x P₃ ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ▷ obj Φ.presheafFiber P₃ ≫
      Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.presheafFiberDesc
        (fun (X : C) (x : Φ.fiber.obj X) ↦
          Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
        ⋯ ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
          obj Φ.presheafFiber P₃ ≫
        Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))

Tactic:
simp only [toPresheafFiber_presheafFiberDesc_assoc, tensorHom_def'_assoc,
        ← comp_whiskerRight_assoc, toPresheafFiber_presheafFiberDesc]

Premises used: CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc, CategoryTheory.MonoidalCategory.tensorHom_def'_assoc, CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc_assoc
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
rw :
  Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
| tensorObj.{v', u'} (𝒞 := inst_1) (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) ◁ Φ.toPresheafFiber X x P₃ ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ▷ obj Φ.presheafFiber P₃ ≫
      Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃))

Tactic:
rw [← tensorHom_def'_assoc, associator_naturality]

Premises used: CategoryTheory.MonoidalCategory.tensorHom_def'_assoc, CategoryTheory.MonoidalCategory.associator_naturality
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
simp :
  Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.presheafFiberDesc
        (fun (X : C) (x : Φ.fiber.obj X) ↦
          Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₁ P₂) ⊗ₘ Φ.toPresheafFiber X x P₃)
        ⋯ ≫
      Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂) ⋯ ▷
          obj Φ.presheafFiber P₃ ≫
        Iso.hom (α_ (obj Φ.presheafFiber P₁) (obj Φ.presheafFiber P₂) (obj Φ.presheafFiber P₃)))

Tactic:
simp only [toPresheafFiber_presheafFiberDesc_assoc, tensorHom_def'_assoc,
        ← comp_whiskerRight_assoc, toPresheafFiber_presheafFiberDesc]
      rw [← tensorHom_def'_assoc, associator_naturality]

Premises used: CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc, CategoryTheory.MonoidalCategory.tensorHom_def'_assoc, CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc, CategoryTheory.MonoidalCategory.associator_naturality, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc_assoc
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
simp :
  NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
    Φ.toPresheafFiber X x P₁ ▷ tensorObj.{v', u'} (𝒞 := inst_1) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X)) ≫
      obj Φ.presheafFiber P₁ ◁ (Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          ⋯ ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            ⋯)

Tactic:
simp only [toPresheafFiber_naturality_assoc, toPresheafFiber_presheafFiberDesc_assoc,
        tensorHom_def_assoc, ← MonoidalCategory.whiskerLeft_comp,
        toPresheafFiber_presheafFiberDesc]

Premises used: CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_naturality_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc_assoc, CategoryTheory.MonoidalCategory.tensorHom_def_assoc
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
rw :
  NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
| NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
    Φ.toPresheafFiber X x P₁ ▷ tensorObj.{v', u'} (𝒞 := inst_1) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X)) ≫
      obj Φ.presheafFiber P₁ ◁ (Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)

Tactic:
rw [← tensorHom_def]

Premises used: CategoryTheory.MonoidalCategory.tensorHom_def
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
dsimp :
  Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
| NatTrans.app (Iso.hom (α_ P₁ P₂ P₃)) (Opposite.op X) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ P₃ : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x : Φ.fiber.obj X
simp :
  Iso.hom (α_ (P₁.obj (Opposite.op X)) (P₂.obj (Opposite.op X)) (P₃.obj (Opposite.op X))) ≫
    (Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
| CategoryStruct.comp (Y := obj Φ.presheafFiber (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (Φ.toPresheafFiber X x (tensorObj (𝒞 := category) (tensorObj (𝒞 := category) P₁ P₂) P₃))
    (map Φ.presheafFiber (Iso.hom (α_ P₁ P₂ P₃)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x P₁ ⊗ₘ Φ.toPresheafFiber X x (tensorObj (𝒞 := category) P₂ P₃))
          ⋯ ≫
        obj Φ.presheafFiber P₁ ◁
          Φ.presheafFiberDesc (fun (X : C) (x : Φ.fiber.obj X) ↦ Φ.toPresheafFiber X x P₂ ⊗ₘ Φ.toPresheafFiber X x P₃)
            ⋯)

Tactic:
simp only [toPresheafFiber_naturality_assoc, toPresheafFiber_presheafFiberDesc_assoc,
        tensorHom_def_assoc, ← MonoidalCategory.whiskerLeft_comp,
        toPresheafFiber_presheafFiberDesc]
      rw [← tensorHom_def]
      dsimp

Premises used: CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_naturality_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonoidalCategory.tensorHom_def, CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_presheafFiberDesc_assoc, CategoryTheory.MonoidalCategory.tensorHom_def_assoc
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
x : Functor.{v, v', u, u'} Cᵒᵖ A
h :
  ∀ (X : C) (x_1 : Φ.fiber.obj X),
    Eq (α :=
      x.obj (Opposite.op X) ⟶
        tensorObj.{v', u'} (𝒞 := inst_1) (tensorUnit.{v', u'} A (𝒞 := inst_1)) (obj Φ.presheafFiber x))
      (Φ.toPresheafFiber X x_1 x ≫ Iso.inv (λ_ (obj Φ.presheafFiber x)))
      (CategoryStruct.comp (Y := obj Φ.presheafFiber x) (Φ.toPresheafFiber X x_1 x)
        (map Φ.presheafFiber (Iso.inv (λ_ x)) ≫
          Φ.presheafFiberDesc
              (fun (X : C) (x_2 : Φ.fiber.obj X) ↦
                Φ.toPresheafFiber X x_2 ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) ⊗ₘ
                  Φ.toPresheafFiber X x_2 x)
              ⋯ ≫
            Φ.presheafFiberDesc
                (fun (x : C) (x_2 : Φ.fiber.obj x) ↦
                  𝟙 (obj.{v, v', u, u'} ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) (Opposite.op x)))
                ⋯ ▷
              obj Φ.presheafFiber x))
⊢ Eq (α :=
    obj Φ.presheafFiber x ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (tensorUnit.{v', u'} A (𝒞 := inst_1)) (obj Φ.presheafFiber x))
    (Iso.inv (λ_ (obj Φ.presheafFiber x)))
    (map Φ.presheafFiber (Iso.inv (λ_ x)) ≫
      Φ.presheafFiberDesc
          (fun (X : C) (x_1 : Φ.fiber.obj X) ↦
            Φ.toPresheafFiber X x_1 ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) ⊗ₘ Φ.toPresheafFiber X x_1 x)
          ⋯ ≫
        Φ.presheafFiberDesc
            (fun (x : C) (x_1 : Φ.fiber.obj x) ↦
              𝟙 (obj.{v, v', u, u'} ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) (Opposite.op x)))
            ⋯ ▷
          obj Φ.presheafFiber x)

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Sites.Point.Monoidal
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : MonoidalCategory.{v', u'} A
inst_3 : HasColimitsOfSize.{w, w, v', u'} A
x : Functor.{v, v', u, u'} Cᵒᵖ A
X : C
x_1 : Φ.fiber.obj X
⊢ Eq (α :=
    x.obj (Opposite.op X) ⟶
      tensorObj.{v', u'} (𝒞 := inst_1) (tensorUnit.{v', u'} A (𝒞 := inst_1)) (obj Φ.presheafFiber x))
    (Φ.toPresheafFiber X x_1 x ≫ Iso.inv (λ_ (obj Φ.presheafFiber x)))
    (CategoryStruct.comp (Y := obj Φ.presheafFiber x) (Φ.toPresheafFiber X x_1 x)
      (map Φ.presheafFiber (Iso.inv (λ_ x)) ≫
        Φ.presheafFiberDesc
            (fun (X : C) (x_2 : Φ.fiber.obj X) ↦
              Φ.toPresheafFiber X x_2 ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) ⊗ₘ
                Φ.toPresheafFiber X x_2 x)
            ⋯ ≫
          Φ.presheafFiberDesc
              (fun (x : C) (x_2 : Φ.fiber.obj x) ↦
                𝟙 (obj.{v, v', u, u'} ((𝟙_ (Functor.{v, v', u, u'} Cᵒᵖ A)) (𝒞 := category)) (Opposite.op x)))
              ⋯ ▷
            obj Φ.presheafFiber x))

Tactic:
simp [tensorHom_def', ← comp_whiskerRight]

Premises used: CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.MonoidalCategory.tensorHom_def'