l3lab/ntp-mathlib huggingface dataset:
instance categoryFreeMonoidalCategory : Category.{u} (F C) where
  Hom X Y := Quotient (FreeMonoidalCategory.setoidHom X Y)
  id X := ⟦FreeMonoidalCategory.Hom.id _⟧
  comp := @fun X Y Z f g =>
    Quotient.map₂ Hom.comp
      (by
        intro f f' hf g g' hg
        exact comp hf hg)
      f g
  id_comp := by
    rintro X Y ⟨f⟩
    exact Quotient.sound (id_comp f)
  comp_id := by
    rintro X Y ⟨f⟩
    exact Quotient.sound (comp_id f)
  assoc := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩ ⟨h⟩
    

Proof state:
case mk.mk.mk
C : Type u
W X Y Z : F C
f✝ : W ⟶ X
f : W ⟶ᵐ X
g✝ : X ⟶ Y
g : X ⟶ᵐ Y
h✝ : Y ⟶ Z
h : Y ⟶ᵐ Z
⊢ (Quot.mk Setoid.r f ≫ Quot.mk Setoid.r g) ≫ Quot.mk Setoid.r h =
    Quot.mk Setoid.r f ≫ Quot.mk Setoid.r g ≫ Quot.mk Setoid.r h

Next tactic:
exact Quotient.sound (assoc f g h)
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      

Proof state:
C : Type u
X₁ Y₁ X₂ Y₂ : F C
⊢ ((fun x x_1 => x ≈ x_1) ⇒ (fun x x_1 => x ≈ x_1) ⇒ fun x x_1 => x ≈ x_1) Hom.tensor Hom.tensor

Next tactic:
intro _ _ h _ _ h'
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      

Proof state:
C : Type u
X₁ Y₁ X₂ Y₂ : F C
a✝¹ b✝¹ : X₁ ⟶ᵐ Y₁
h : a✝¹ ≈ b✝¹
a✝ b✝ : X₂ ⟶ᵐ Y₂
h' : a✝ ≈ b✝
⊢ Hom.tensor a✝¹ a✝ ≈ Hom.tensor b✝¹ b✝

Next tactic:
exact HomEquiv.tensor h h'
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    

Proof state:
C : Type u
⊢ ∀ {X₁ Y₁ X₂ Y₂ : F C} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂), f ⊗ g = f ▷ X₂ ≫ Y₁ ◁ g

Next tactic:
rintro W X Y Z ⟨f⟩ ⟨g⟩
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    

Proof state:
case mk.mk
C : Type u
W X Y Z : F C
f✝ : W ⟶ X
f : W ⟶ᵐ X
g✝ : Y ⟶ Z
g : Y ⟶ᵐ Z
⊢ Quot.mk Setoid.r f ⊗ Quot.mk Setoid.r g = Quot.mk Setoid.r f ▷ Y ≫ X ◁ Quot.mk Setoid.r g

Next tactic:
apply Quotient.sound
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    

Proof state:
case mk.mk.a
C : Type u
W X Y Z : F C
f✝ : W ⟶ X
f : W ⟶ᵐ X
g✝ : Y ⟶ Z
g : Y ⟶ᵐ Z
⊢ Hom.tensor f g ≈ Hom.comp ((fun f' => Hom.tensor f' (Hom.id Y)) f) ((fun f' => Hom.tensor (Hom.id X) f') g)

Next tactic:
calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        

Proof state:
C : Type u
W X Y Z : F C
f✝ : W ⟶ X
f : W ⟶ᵐ X
g✝ : Y ⟶ Z
g : Y ⟶ᵐ Z
⊢ Hom.tensor f g ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g)

Next tactic:
apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        

Proof state:
C : Type u
W X Y Z : F C
f✝ : W ⟶ X
f : W ⟶ᵐ X
g✝ : Y ⟶ Z
g : Y ⟶ᵐ Z
⊢ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) ≈
    Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g)

Next tactic:
apply HomEquiv.tensor_comp
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  tensor_id X Y := Quotient.sound tensor_id
  tensor_comp := @fun X₁ Y₁ Z₁ X₂ Y₂ Z₂ => by
    

Proof state:
C : Type u
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : F C
⊢ ∀ (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂), f₁ ≫ g₁ ⊗ f₂ ≫ g₂ = (f₁ ⊗ f₂) ≫ (g₁ ⊗ g₂)

Next tactic:
rintro ⟨f₁⟩ ⟨f₂⟩ ⟨g₁⟩ ⟨g₂⟩
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  tensor_id X Y := Quotient.sound tensor_id
  tensor_comp := @fun X₁ Y₁ Z₁ X₂ Y₂ Z₂ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨g₁⟩ ⟨g₂⟩
    

Proof state:
case mk.mk.mk.mk
C : Type u
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : F C
f₁✝ : X₁ ⟶ Y₁
f₁ : X₁ ⟶ᵐ Y₁
f₂✝ : X₂ ⟶ Y₂
f₂ : X₂ ⟶ᵐ Y₂
g₁✝ : Y₁ ⟶ Z₁
g₁ : Y₁ ⟶ᵐ Z₁
g₂✝ : Y₂ ⟶ Z₂
g₂ : Y₂ ⟶ᵐ Z₂
⊢ Quot.mk Setoid.r f₁ ≫ Quot.mk Setoid.r g₁ ⊗ Quot.mk Setoid.r f₂ ≫ Quot.mk Setoid.r g₂ =
    (Quot.mk Setoid.r f₁ ⊗ Quot.mk Setoid.r f₂) ≫ (Quot.mk Setoid.r g₁ ⊗ Quot.mk Setoid.r g₂)

Next tactic:
exact Quotient.sound (tensor_comp _ _ _ _)
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    

Proof state:
C : Type u
⊢ ∀ (X Y : F C), X ◁ 𝟙 Y = 𝟙 (X ⊗ Y)

Next tactic:
rintro X Y
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    

Proof state:
C : Type u
X Y : F C
⊢ X ◁ 𝟙 Y = 𝟙 (X ⊗ Y)

Next tactic:
apply Quotient.sound
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    

Proof state:
case a
C : Type u
X Y : F C
⊢ (fun f' => Hom.tensor (Hom.id X) f') (Hom.id Y) ≈ Hom.id (X ⊗ Y)

Next tactic:
apply HomEquiv.tensor_id
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    

Proof state:
C : Type u
⊢ ∀ (X Y : F C), 𝟙 X ▷ Y = 𝟙 (X ⊗ Y)

Next tactic:
intro X Y
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    

Proof state:
C : Type u
X Y : F C
⊢ 𝟙 X ▷ Y = 𝟙 (X ⊗ Y)

Next tactic:
apply Quotient.sound
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    apply Quotient.sound
    

Proof state:
case a
C : Type u
X Y : F C
⊢ (fun f' => Hom.tensor f' (Hom.id Y)) (Hom.id X) ≈ Hom.id (X ⊗ Y)

Next tactic:
apply HomEquiv.tensor_id
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  tensor_id X Y := Quotient.sound tensor_id
  tensor_comp := @fun X₁ Y₁ Z₁ X₂ Y₂ Z₂ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨g₁⟩ ⟨g₂⟩
    exact Quotient.sound (tensor_comp _ _ _ _)
  tensorUnit := FreeMonoidalCategory.unit
  associator X Y Z :=
    ⟨⟦Hom.α_hom X Y Z⟧, ⟦Hom.α_inv X Y Z⟧, Quotient.sound α_hom_inv, Quotient.sound α_inv_hom⟩
  associator_naturality := @fun X₁ X₂ X₃ Y₁ Y₂ Y₃ => by
    

Proof state:
C : Type u
X₁ X₂ X₃ Y₁ Y₂ Y₃ : F C
⊢ ∀ (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃),
    ((f₁ ⊗ f₂) ⊗ f₃) ≫ (α_ Y₁ Y₂ Y₃).hom = (α_ X₁ X₂ X₃).hom ≫ (f₁ ⊗ f₂ ⊗ f₃)

Next tactic:
rintro ⟨f₁⟩ ⟨f₂⟩ ⟨f₃⟩
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  tensor_id X Y := Quotient.sound tensor_id
  tensor_comp := @fun X₁ Y₁ Z₁ X₂ Y₂ Z₂ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨g₁⟩ ⟨g₂⟩
    exact Quotient.sound (tensor_comp _ _ _ _)
  tensorUnit := FreeMonoidalCategory.unit
  associator X Y Z :=
    ⟨⟦Hom.α_hom X Y Z⟧, ⟦Hom.α_inv X Y Z⟧, Quotient.sound α_hom_inv, Quotient.sound α_inv_hom⟩
  associator_naturality := @fun X₁ X₂ X₃ Y₁ Y₂ Y₃ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨f₃⟩
    

Proof state:
case mk.mk.mk
C : Type u
X₁ X₂ X₃ Y₁ Y₂ Y₃ : F C
f₁✝ : X₁ ⟶ Y₁
f₁ : X₁ ⟶ᵐ Y₁
f₂✝ : X₂ ⟶ Y₂
f₂ : X₂ ⟶ᵐ Y₂
f₃✝ : X₃ ⟶ Y₃
f₃ : X₃ ⟶ᵐ Y₃
⊢ ((Quot.mk Setoid.r f₁ ⊗ Quot.mk Setoid.r f₂) ⊗ Quot.mk Setoid.r f₃) ≫ (α_ Y₁ Y₂ Y₃).hom =
    (α_ X₁ X₂ X₃).hom ≫ (Quot.mk Setoid.r f₁ ⊗ Quot.mk Setoid.r f₂ ⊗ Quot.mk Setoid.r f₃)

Next tactic:
exact Quotient.sound (associator_naturality _ _ _)
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  tensor_id X Y := Quotient.sound tensor_id
  tensor_comp := @fun X₁ Y₁ Z₁ X₂ Y₂ Z₂ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨g₁⟩ ⟨g₂⟩
    exact Quotient.sound (tensor_comp _ _ _ _)
  tensorUnit := FreeMonoidalCategory.unit
  associator X Y Z :=
    ⟨⟦Hom.α_hom X Y Z⟧, ⟦Hom.α_inv X Y Z⟧, Quotient.sound α_hom_inv, Quotient.sound α_inv_hom⟩
  associator_naturality := @fun X₁ X₂ X₃ Y₁ Y₂ Y₃ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨f₃⟩
    exact Quotient.sound (associator_naturality _ _ _)
  leftUnitor X := ⟨⟦Hom.l_hom X⟧, ⟦Hom.l_inv X⟧, Quotient.sound l_hom_inv, Quotient.sound l_inv_hom⟩
  leftUnitor_naturality := @fun X Y => by
    

Proof state:
C : Type u
X Y : F C
⊢ ∀ (f : X ⟶ Y), (𝟙 (𝟙_ (F C)) ⊗ f) ≫ (λ_ Y).hom = (λ_ X).hom ≫ f

Next tactic:
rintro ⟨f⟩
instance : MonoidalCategory (F C) where
  tensorObj X Y := FreeMonoidalCategory.tensor X Y
  tensorHom := @fun X₁ Y₁ X₂ Y₂ =>
    Quotient.map₂ Hom.tensor <| by
      intro _ _ h _ _ h'
      exact HomEquiv.tensor h h'
  whiskerLeft := fun X _ _ f =>
    Quotient.map (fun f' => Hom.tensor (Hom.id X) f')
      (fun _ _ h => HomEquiv.tensor (HomEquiv.refl (Hom.id X)) h) f
  whiskerRight := fun f Y =>
    Quotient.map (fun f' => Hom.tensor f' (Hom.id Y))
      (fun _ _ h => HomEquiv.tensor h (HomEquiv.refl (Hom.id Y))) f
  tensorHom_def := by
    rintro W X Y Z ⟨f⟩ ⟨g⟩
    apply Quotient.sound
    calc Hom.tensor f g
      _ ≈ Hom.tensor (Hom.comp f (Hom.id X)) (Hom.comp (Hom.id Y) g) := by
        apply HomEquiv.tensor (HomEquiv.comp_id f).symm (HomEquiv.id_comp g).symm
      _ ≈ Hom.comp (Hom.tensor f (Hom.id Y)) (Hom.tensor (Hom.id X) g) := by
        apply HomEquiv.tensor_comp
  whiskerLeft_id := by
    rintro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  id_whiskerRight := by
    intro X Y
    apply Quotient.sound
    apply HomEquiv.tensor_id
  tensor_id X Y := Quotient.sound tensor_id
  tensor_comp := @fun X₁ Y₁ Z₁ X₂ Y₂ Z₂ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨g₁⟩ ⟨g₂⟩
    exact Quotient.sound (tensor_comp _ _ _ _)
  tensorUnit := FreeMonoidalCategory.unit
  associator X Y Z :=
    ⟨⟦Hom.α_hom X Y Z⟧, ⟦Hom.α_inv X Y Z⟧, Quotient.sound α_hom_inv, Quotient.sound α_inv_hom⟩
  associator_naturality := @fun X₁ X₂ X₃ Y₁ Y₂ Y₃ => by
    rintro ⟨f₁⟩ ⟨f₂⟩ ⟨f₃⟩
    exact Quotient.sound (associator_naturality _ _ _)
  leftUnitor X := ⟨⟦Hom.l_hom X⟧, ⟦Hom.l_inv X⟧, Quotient.sound l_hom_inv, Quotient.sound l_inv_hom⟩
  leftUnitor_naturality := @fun X Y => by
    rintro ⟨f⟩
    

Proof state:
case mk
C : Type u
X Y : F C
f✝ : X ⟶ Y
f : X ⟶ᵐ Y
⊢ (𝟙 (𝟙_ (F C)) ⊗ Quot.mk Setoid.r f) ≫ (λ_ Y).hom = (λ_ X).hom ≫ Quot.mk Setoid.r f

Next tactic:
exact Quotient.sound (l_naturality _)