awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.LaxMonoidal) toLaxMonoidal toLaxMonoidal
x x_1 : C
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (Iso.hom (μIso F x x_1)) (δ F x x_1))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (μ F x x_1) (δ F x x_1))

Tactic:
exact congr(($eq.symm).μ _ _ ≫ _)
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
ε :
  Eq (α := tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ⟶ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F)
    (LaxMonoidal.ε F)
μ :
  Eq (α :=
    (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ⟶ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y))
    (LaxMonoidal.μ F) (LaxMonoidal.μ F)
η :
  Eq (α := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (OplaxMonoidal.η F)
    (OplaxMonoidal.η F)
δ :
  Eq (α :=
    (X Y : C) → F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y))
    (OplaxMonoidal.δ F) (OplaxMonoidal.δ F)
⊢ a = b

Tactic:
ext1
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
ε :
  Eq (α := tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F) (Iso.inv (εIso F)))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F) (Iso.inv (εIso F)))
⊢ Eq (α := tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ⟶ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F)
    (LaxMonoidal.ε F)

Tactic:
rw [← cancel_mono (εIso _).inv]

Premises used: CategoryTheory.cancel_mono, CategoryTheory.Functor.Monoidal.εIso, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
ε :
  Eq (α := tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F) (Iso.inv (εIso F)))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F) (η F))
⊢ Eq (α := tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F) (Iso.inv (εIso F)))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (LaxMonoidal.ε F) (Iso.inv (εIso F)))

Tactic:
rw [εIso_inv, ε_η, ← @ε_η _ _ _ _ _ _ _ a, ← εIso_inv]

Premises used: CategoryTheory.Functor.Monoidal.ε_η, CategoryTheory.Functor.Monoidal.εIso_inv
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
⊢ Eq (α := tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (ε F) (Iso.inv (εIso F)))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (ε F) (η F))

Tactic:
congr(_ ≫ ($eq.symm).η)

Premises used: congrArg, CategoryTheory.Functor.LaxMonoidal.ε, id, CategoryTheory.CategoryStruct.comp, Eq.symm, CategoryTheory.Functor.OplaxMonoidal.η
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
⊢ Eq (α := tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (ε F) (Iso.inv (εIso F)))
    (CategoryStruct.comp (Y := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (ε F) (η F))

Tactic:
exact congr(_ ≫ ($eq.symm).η)
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
μ :
  ∀ (x x_1 : C),
    Eq (α := tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1))
      (LaxMonoidal.μ F x x_1) (LaxMonoidal.μ F x x_1)
⊢ Eq (α :=
    (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ⟶ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y))
    (LaxMonoidal.μ F) (LaxMonoidal.μ F)

Tactic:
ext
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
x x_1 : C
μ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (LaxMonoidal.μ F x x_1)
      (Iso.inv (μIso F x x_1)))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (LaxMonoidal.μ F x x_1)
      (Iso.inv (μIso F x x_1)))
⊢ Eq (α := tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1))
    (LaxMonoidal.μ F x x_1) (LaxMonoidal.μ F x x_1)

Tactic:
rw [← cancel_mono (μIso F _ _).inv]

Premises used: CategoryTheory.cancel_mono, CategoryTheory.Functor.Monoidal.μIso, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
x x_1 : C
μ :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (LaxMonoidal.μ F x x_1)
      (Iso.inv (μIso F x x_1)))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (LaxMonoidal.μ F x x_1) (δ F x x_1))
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (LaxMonoidal.μ F x x_1)
      (Iso.inv (μIso F x x_1)))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (LaxMonoidal.μ F x x_1)
      (Iso.inv (μIso F x x_1)))

Tactic:
rw [μIso_inv, μ_δ, ← @μ_δ _ _ _ _ _ _ _ a, ← μIso_inv]

Premises used: CategoryTheory.Functor.Monoidal.μIso_inv, CategoryTheory.Functor.Monoidal.μ_δ
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
x x_1 : C
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (μ F x x_1) (Iso.inv (μIso F x x_1)))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (μ F x x_1) (δ F x x_1))

Tactic:
congr(_ ≫ ($eq.symm).δ _ _)

Premises used: congrArg, CategoryTheory.Functor.OplaxMonoidal.δ, id, CategoryTheory.Functor.LaxMonoidal.μ, CategoryTheory.CategoryStruct.comp, Eq.symm
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
x x_1 : C
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj x) (F.obj x_1))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (μ F x x_1) (Iso.inv (μIso F x x_1)))
    (CategoryStruct.comp (Y := F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x x_1)) (μ F x x_1) (δ F x x_1))

Tactic:
exact congr(_ ≫ ($eq.symm).δ _ _)
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
⊢ Eq (α := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (η F) (η F)

Tactic:
congr(($eq).η)

Premises used: congrArg, id, CategoryTheory.Functor.OplaxMonoidal.η
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
⊢ Eq (α := F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶ tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (η F) (η F)

Tactic:
exact congr(($eq).η)
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
⊢ Eq (α :=
    (X Y : C) → F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (δ F)
    (δ F)

Tactic:
congr(($eq).δ)

Premises used: congrArg, CategoryTheory.Functor.OplaxMonoidal.δ, id
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
a b : F.Monoidal
eq : Eq (α := F.OplaxMonoidal) toOplaxMonoidal toOplaxMonoidal
⊢ Eq (α :=
    (X Y : C) → F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) ⟶ tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) (δ F)
    (δ F)

Tactic:
exact congr(($eq).δ)
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
E : Type u₃
inst_4 : Category.{v₃, u₃} E
inst_5 : MonoidalCategory.{v₃, u₃} E
C' : Type u₁'
inst_6 : Category.{v₁', u₁'} C'
F : Functor.{v₁, v₂, u₁, u₂} C D
εIso : tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ≅ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))
μIso : (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ≅ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y)
μIso_inv_natural_left :
  ∀ {X Y : C} (f : X ⟶ Y) (X' : C), (μIso X X').inv ≫ F.map f ▷ F.obj X' = F.map (f ▷ X') ≫ (μIso Y X').inv
μIso_inv_natural_right :
  ∀ {X Y : C} (X' : C) (f : X ⟶ Y), (μIso X' X).inv ≫ F.obj X' ◁ F.map f = F.map (X' ◁ f) ≫ (μIso X' Y).inv
oplax_associativity :
  ∀ (X Y Z : C),
    Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
        (μIso X Y).inv ▷ F.obj Z ≫ Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) =
      F.map (Iso.hom (α_ X Y Z)) ≫ Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫ F.obj X ◁ (μIso Y Z).inv
oplax_left_unitality :
  ∀ (X : C),
    Iso.inv (λ_ (F.obj X)) =
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X
oplax_right_unitality :
  ∀ (X : C),
    Iso.inv (ρ_ (F.obj X)) =
      F.map (Iso.inv (ρ_ X)) ≫ Iso.inv (μIso X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ F.obj X ◁ εIso.inv
X Y : C
f : X ⟶ Y
X' : C
⊢ F.map f ▷ F.obj X' ≫ (μIso Y X').hom = (μIso X X').hom ≫ F.map (f ▷ X')

Tactic:
simp [← cancel_epi (μIso X X').inv, reassoc_of% μIso_inv_natural_left f X']

Premises used: CategoryTheory.cancel_epi, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
E : Type u₃
inst_4 : Category.{v₃, u₃} E
inst_5 : MonoidalCategory.{v₃, u₃} E
C' : Type u₁'
inst_6 : Category.{v₁', u₁'} C'
F : Functor.{v₁, v₂, u₁, u₂} C D
εIso : tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ≅ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))
μIso : (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ≅ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y)
μIso_inv_natural_left :
  ∀ {X Y : C} (f : X ⟶ Y) (X' : C), (μIso X X').inv ≫ F.map f ▷ F.obj X' = F.map (f ▷ X') ≫ (μIso Y X').inv
μIso_inv_natural_right :
  ∀ {X Y : C} (X' : C) (f : X ⟶ Y), (μIso X' X).inv ≫ F.obj X' ◁ F.map f = F.map (X' ◁ f) ≫ (μIso X' Y).inv
oplax_associativity :
  ∀ (X Y Z : C),
    Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
        (μIso X Y).inv ▷ F.obj Z ≫ Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) =
      F.map (Iso.hom (α_ X Y Z)) ≫ Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫ F.obj X ◁ (μIso Y Z).inv
oplax_left_unitality :
  ∀ (X : C),
    Iso.inv (λ_ (F.obj X)) =
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X
oplax_right_unitality :
  ∀ (X : C),
    Iso.inv (ρ_ (F.obj X)) =
      F.map (Iso.inv (ρ_ X)) ≫ Iso.inv (μIso X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ F.obj X ◁ εIso.inv
X Y X' : C
g : X ⟶ Y
⊢ F.obj X' ◁ F.map g ≫ (μIso X' Y).hom = (μIso X' X).hom ≫ F.map (X' ◁ g)

Tactic:
simp [← cancel_mono (μIso X' Y).inv, ← (μIso_inv_natural_right X' g)]

Premises used: CategoryTheory.cancel_mono, CategoryTheory.Iso.inv
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
E : Type u₃
inst_4 : Category.{v₃, u₃} E
inst_5 : MonoidalCategory.{v₃, u₃} E
C' : Type u₁'
inst_6 : Category.{v₁', u₁'} C'
F : Functor.{v₁, v₂, u₁, u₂} C D
εIso : tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ≅ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))
μIso : (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ≅ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y)
μIso_inv_natural_left :
  ∀ {X Y : C} (f : X ⟶ Y) (X' : C), (μIso X X').inv ≫ F.map f ▷ F.obj X' = F.map (f ▷ X') ≫ (μIso Y X').inv
μIso_inv_natural_right :
  ∀ {X Y : C} (X' : C) (f : X ⟶ Y), (μIso X' X).inv ≫ F.obj X' ◁ F.map f = F.map (X' ◁ f) ≫ (μIso X' Y).inv
oplax_associativity :
  ∀ (X Y Z : C),
    Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
        (μIso X Y).inv ▷ F.obj Z ≫ Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) =
      F.map (Iso.hom (α_ X Y Z)) ≫ Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫ F.obj X ◁ (μIso Y Z).inv
oplax_left_unitality :
  ∀ (X : C),
    Iso.inv (λ_ (F.obj X)) =
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X
oplax_right_unitality :
  ∀ (X : C),
    Iso.inv (ρ_ (F.obj X)) =
      F.map (Iso.inv (ρ_ X)) ≫ Iso.inv (μIso X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ F.obj X ◁ εIso.inv
X Y Z : C
rw :
  Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
      (μIso X Y).inv ▷ F.obj Z ≫
        (μIso X Y).hom ▷ F.obj Z ≫ Iso.hom (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫ F.map (Iso.hom (α_ X Y Z)) =
    F.map (Iso.hom (α_ X Y Z)) ≫
      Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫
        F.obj X ◁ (μIso Y Z).inv ≫ F.obj X ◁ (μIso Y Z).hom ≫ Iso.hom (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z))
⊢ (μIso X Y).hom ▷ F.obj Z ≫ Iso.hom (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫ F.map (Iso.hom (α_ X Y Z)) =
    Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) ≫
      F.obj X ◁ (μIso Y Z).hom ≫ Iso.hom (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z))

Tactic:
rw [← cancel_epi ((μIso X Y).inv ▷ F.obj Z), ← cancel_epi (μIso (X ⊗ Y) Z).inv,
      reassoc_of% oplax_associativity]

Premises used: CategoryTheory.MonoidalCategoryStruct.whiskerRight, CategoryTheory.MonoidalCategoryStruct.tensorObj, CategoryTheory.cancel_epi, CategoryTheory.Iso.inv, CategoryTheory.Functor.obj
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
E : Type u₃
inst_4 : Category.{v₃, u₃} E
inst_5 : MonoidalCategory.{v₃, u₃} E
C' : Type u₁'
inst_6 : Category.{v₁', u₁'} C'
F : Functor.{v₁, v₂, u₁, u₂} C D
εIso : tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ≅ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))
μIso : (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ≅ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y)
μIso_inv_natural_left :
  ∀ {X Y : C} (f : X ⟶ Y) (X' : C), (μIso X X').inv ≫ F.map f ▷ F.obj X' = F.map (f ▷ X') ≫ (μIso Y X').inv
μIso_inv_natural_right :
  ∀ {X Y : C} (X' : C) (f : X ⟶ Y), (μIso X' X).inv ≫ F.obj X' ◁ F.map f = F.map (X' ◁ f) ≫ (μIso X' Y).inv
oplax_associativity :
  ∀ (X Y Z : C),
    Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
        (μIso X Y).inv ▷ F.obj Z ≫ Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) =
      F.map (Iso.hom (α_ X Y Z)) ≫ Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫ F.obj X ◁ (μIso Y Z).inv
oplax_left_unitality :
  ∀ (X : C),
    Iso.inv (λ_ (F.obj X)) =
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X
oplax_right_unitality :
  ∀ (X : C),
    Iso.inv (ρ_ (F.obj X)) =
      F.map (Iso.inv (ρ_ X)) ≫ Iso.inv (μIso X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ F.obj X ◁ εIso.inv
X Y Z : C
⊢ Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
      (μIso X Y).inv ▷ F.obj Z ≫
        (μIso X Y).hom ▷ F.obj Z ≫ Iso.hom (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫ F.map (Iso.hom (α_ X Y Z)) =
    F.map (Iso.hom (α_ X Y Z)) ≫
      Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫
        F.obj X ◁ (μIso Y Z).inv ≫ F.obj X ◁ (μIso Y Z).hom ≫ Iso.hom (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
E : Type u₃
inst_4 : Category.{v₃, u₃} E
inst_5 : MonoidalCategory.{v₃, u₃} E
C' : Type u₁'
inst_6 : Category.{v₁', u₁'} C'
F : Functor.{v₁, v₂, u₁, u₂} C D
εIso : tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ≅ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))
μIso : (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ≅ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y)
μIso_inv_natural_left :
  ∀ {X Y : C} (f : X ⟶ Y) (X' : C), (μIso X X').inv ≫ F.map f ▷ F.obj X' = F.map (f ▷ X') ≫ (μIso Y X').inv
μIso_inv_natural_right :
  ∀ {X Y : C} (X' : C) (f : X ⟶ Y), (μIso X' X).inv ≫ F.obj X' ◁ F.map f = F.map (X' ◁ f) ≫ (μIso X' Y).inv
oplax_associativity :
  ∀ (X Y Z : C),
    Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
        (μIso X Y).inv ▷ F.obj Z ≫ Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) =
      F.map (Iso.hom (α_ X Y Z)) ≫ Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫ F.obj X ◁ (μIso Y Z).inv
oplax_left_unitality :
  ∀ (X : C),
    Iso.inv (λ_ (F.obj X)) =
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X
oplax_right_unitality :
  ∀ (X : C),
    Iso.inv (ρ_ (F.obj X)) =
      F.map (Iso.inv (ρ_ X)) ≫ Iso.inv (μIso X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ F.obj X ◁ εIso.inv
X : C
rw :
  𝟙 (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj X)) =
    (εIso.hom ▷ F.obj X ≫ Iso.hom (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ F.map (Iso.hom (λ_ X))) ≫
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X
⊢ Iso.hom (λ_ (F.obj X)) =
    εIso.hom ▷ F.obj X ≫ Iso.hom (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ F.map (Iso.hom (λ_ X))

Tactic:
rw [← cancel_mono (λ_ (F.obj X)).inv, Iso.hom_inv_id, oplax_left_unitality]

Premises used: CategoryTheory.MonoidalCategoryStruct.leftUnitor, CategoryTheory.Iso.hom_inv_id, CategoryTheory.cancel_mono, CategoryTheory.Iso.inv, CategoryTheory.Functor.obj
-- Module: Mathlib.CategoryTheory.Monoidal.Functor
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
E : Type u₃
inst_4 : Category.{v₃, u₃} E
inst_5 : MonoidalCategory.{v₃, u₃} E
C' : Type u₁'
inst_6 : Category.{v₁', u₁'} C'
F : Functor.{v₁, v₂, u₁, u₂} C D
εIso : tensorUnit.{v₂, u₂} D (𝒞 := inst_2) ≅ F.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))
μIso : (X Y : C) → tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ≅ F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) X Y)
μIso_inv_natural_left :
  ∀ {X Y : C} (f : X ⟶ Y) (X' : C), (μIso X X').inv ≫ F.map f ▷ F.obj X' = F.map (f ▷ X') ≫ (μIso Y X').inv
μIso_inv_natural_right :
  ∀ {X Y : C} (X' : C) (f : X ⟶ Y), (μIso X' X).inv ≫ F.obj X' ◁ F.map f = F.map (X' ◁ f) ≫ (μIso X' Y).inv
oplax_associativity :
  ∀ (X Y Z : C),
    Iso.inv (μIso (tensorObj.{v₁, u₁} (𝒞 := inst) X Y) Z) ≫
        (μIso X Y).inv ▷ F.obj Z ≫ Iso.hom (α_ (F.obj X) (F.obj Y) (F.obj Z)) =
      F.map (Iso.hom (α_ X Y Z)) ≫ Iso.inv (μIso X (tensorObj.{v₁, u₁} (𝒞 := inst) Y Z)) ≫ F.obj X ◁ (μIso Y Z).inv
oplax_left_unitality :
  ∀ (X : C),
    Iso.inv (λ_ (F.obj X)) =
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X
oplax_right_unitality :
  ∀ (X : C),
    Iso.inv (ρ_ (F.obj X)) =
      F.map (Iso.inv (ρ_ X)) ≫ Iso.inv (μIso X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ F.obj X ◁ εIso.inv
X : C
⊢ 𝟙 (tensorObj.{v₂, u₂} (𝒞 := inst_2) (tensorUnit.{v₂, u₂} D (𝒞 := inst_2)) (F.obj X)) =
    (εIso.hom ▷ F.obj X ≫ Iso.hom (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ F.map (Iso.hom (λ_ X))) ≫
      F.map (Iso.inv (λ_ X)) ≫ Iso.inv (μIso (tensorUnit.{v₁, u₁} C (𝒞 := inst)) X) ≫ εIso.inv ▷ F.obj X

Tactic:
simp