l3lab/ntp-mathlib huggingface dataset:
theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
    (g₂ : Y₂ ⟶ Z₂) : tensorHom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom ℬ f₁ f₂ ≫ tensorHom ℬ g₁ g₂ := by
  apply IsLimit.hom_ext (ℬ _ _).isLimit;
  rintro ⟨⟨⟩⟩ <;>
    · dsimp [tensorHom]
      

Proof state:
case mk.left
C : Type u
inst✝ : Category.{v, u} C
X Y : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ IsLimit.lift (ℬ Z₁ Z₂).isLimit
        (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁ ≫ g₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂ ≫ g₂)) ≫
      BinaryFan.fst (ℬ Z₁ Z₂).cone =
    (IsLimit.lift (ℬ Y₁ Y₂).isLimit
          (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)) ≫
        IsLimit.lift (ℬ Z₁ Z₂).isLimit
          (BinaryFan.mk (BinaryFan.fst (ℬ Y₁ Y₂).cone ≫ g₁) (BinaryFan.snd (ℬ Y₁ Y₂).cone ≫ g₂))) ≫
      BinaryFan.fst (ℬ Z₁ Z₂).cone

Next tactic:
simp
theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
    (g₂ : Y₂ ⟶ Z₂) : tensorHom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom ℬ f₁ f₂ ≫ tensorHom ℬ g₁ g₂ := by
  apply IsLimit.hom_ext (ℬ _ _).isLimit;
  rintro ⟨⟨⟩⟩ <;>
    · 

Proof state:
case mk.right
C : Type u
inst✝ : Category.{v, u} C
X Y : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ tensorHom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) ≫ (ℬ Z₁ Z₂).cone.π.app { as := WalkingPair.right } =
    (tensorHom ℬ f₁ f₂ ≫ tensorHom ℬ g₁ g₂) ≫ (ℬ Z₁ Z₂).cone.π.app { as := WalkingPair.right }

Next tactic:
dsimp [tensorHom]
theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
    (g₂ : Y₂ ⟶ Z₂) : tensorHom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom ℬ f₁ f₂ ≫ tensorHom ℬ g₁ g₂ := by
  apply IsLimit.hom_ext (ℬ _ _).isLimit;
  rintro ⟨⟨⟩⟩ <;>
    · dsimp [tensorHom]
      

Proof state:
case mk.right
C : Type u
inst✝ : Category.{v, u} C
X Y : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
g₁ : Y₁ ⟶ Z₁
g₂ : Y₂ ⟶ Z₂
⊢ IsLimit.lift (ℬ Z₁ Z₂).isLimit
        (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁ ≫ g₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂ ≫ g₂)) ≫
      BinaryFan.snd (ℬ Z₁ Z₂).cone =
    (IsLimit.lift (ℬ Y₁ Y₂).isLimit
          (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)) ≫
        IsLimit.lift (ℬ Z₁ Z₂).isLimit
          (BinaryFan.mk (BinaryFan.fst (ℬ Y₁ Y₂).cone ≫ g₁) (BinaryFan.snd (ℬ Y₁ Y₂).cone ≫ g₂))) ≫
      BinaryFan.snd (ℬ Z₁ Z₂).cone

Next tactic:
simp
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
      (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
        tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
    (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
      (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom

Next tactic:
dsimp [tensorHom]
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
        (BinaryFan.mk
          (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
          (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
      (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
        IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
          (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
            (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
              (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom)) =
    (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
      (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom

Next tactic:
apply IsLimit.hom_ext (ℬ _ _).isLimit
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ ∀ (j : Discrete WalkingPair),
    (IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
            (BinaryFan.mk
              (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
              (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
          (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
            IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
              (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
                (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                  (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
        (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app j =
      ((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
          (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
        (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app j

Next tactic:
rintro ⟨⟨⟩⟩
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · 

Proof state:
case mk.left
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ (IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
          (BinaryFan.mk
            (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
            (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
            (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
              (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
      (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.left } =
    ((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
      (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.left }

Next tactic:
simp
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · simp
  · 

Proof state:
case mk.right
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ (IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
          (BinaryFan.mk
            (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
            (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
            (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
              (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
      (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right } =
    ((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
      (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }

Next tactic:
apply IsLimit.hom_ext (ℬ _ _).isLimit
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · simp
  · apply IsLimit.hom_ext (ℬ _ _).isLimit
    

Proof state:
case mk.right
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ ∀ (j : Discrete WalkingPair),
    ((IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
              (BinaryFan.mk
                (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
                (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
            (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
              IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
                (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
                  (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                    (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
          (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ X (ℬ Y Z).cone.pt).cone.π.app j =
      (((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
            (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
          (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ X (ℬ Y Z).cone.pt).cone.π.app j

Next tactic:
rintro ⟨⟨⟩⟩
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · simp
  · apply IsLimit.hom_ext (ℬ _ _).isLimit
    rintro ⟨⟨⟩⟩
    · 

Proof state:
case mk.right.mk.left
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ ((IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
            (BinaryFan.mk
              (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
              (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
          (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
            IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
              (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
                (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                  (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
        (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.left } =
    (((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
          (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
        (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.left }

Next tactic:
simp
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · simp
  · apply IsLimit.hom_ext (ℬ _ _).isLimit
    rintro ⟨⟨⟩⟩
    · simp
    

Proof state:
case mk.right.mk.right
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ ((IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
            (BinaryFan.mk
              (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
              (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
          (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
            IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
              (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
                (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                  (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
        (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right } =
    (((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
          (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
        (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right }

Next tactic:
apply IsLimit.hom_ext (ℬ _ _).isLimit
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · simp
  · apply IsLimit.hom_ext (ℬ _ _).isLimit
    rintro ⟨⟨⟩⟩
    · simp
    apply IsLimit.hom_ext (ℬ _ _).isLimit
    

Proof state:
case mk.right.mk.right
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ ∀ (j : Discrete WalkingPair),
    (((IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
                (BinaryFan.mk
                  (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫
                    (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
                  (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
              (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
                IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
                  (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
                    (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                      (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
            (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
          (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ Y Z).cone.π.app j =
      ((((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
              (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
            (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
          (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ Y Z).cone.π.app j

Next tactic:
rintro ⟨⟨⟩⟩
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · simp
  · apply IsLimit.hom_ext (ℬ _ _).isLimit
    rintro ⟨⟨⟩⟩
    · simp
    apply IsLimit.hom_ext (ℬ _ _).isLimit
    rintro ⟨⟨⟩⟩
    · 

Proof state:
case mk.right.mk.right.mk.left
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ (((IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
              (BinaryFan.mk
                (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
                (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
            (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
              IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
                (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
                  (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                    (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
          (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ Y Z).cone.π.app { as := WalkingPair.left } =
    ((((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
            (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
          (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ Y Z).cone.π.app { as := WalkingPair.left }

Next tactic:
simp
theorem pentagon (W X Y Z : C) :
    tensorHom ℬ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom (𝟙 Z) ≫
        (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
          tensorHom ℬ (𝟙 W) (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom =
      (BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
        (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; rintro ⟨⟨⟩⟩
  · simp
  · apply IsLimit.hom_ext (ℬ _ _).isLimit
    rintro ⟨⟨⟩⟩
    · simp
    apply IsLimit.hom_ext (ℬ _ _).isLimit
    rintro ⟨⟨⟩⟩
    · simp
    · 

Proof state:
case mk.right.mk.right.mk.right
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
W X Y Z : C
⊢ (((IsLimit.lift (ℬ (ℬ W (ℬ X Y).cone.pt).cone.pt Z).isLimit
              (BinaryFan.mk
                (BinaryFan.fst (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ (BinaryFan.associatorOfLimitCone ℬ W X Y).hom)
                (BinaryFan.snd (ℬ (ℬ (ℬ W X).cone.pt Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
            (BinaryFan.associatorOfLimitCone ℬ W (tensorObj ℬ X Y) Z).hom ≫
              IsLimit.lift (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).isLimit
                (BinaryFan.mk (BinaryFan.fst (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫ 𝟙 W)
                  (BinaryFan.snd (ℬ W (ℬ (tensorObj ℬ X Y) Z).cone.pt).cone ≫
                    (BinaryFan.associatorOfLimitCone ℬ X Y Z).hom))) ≫
          (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ Y Z).cone.π.app { as := WalkingPair.right } =
    ((((BinaryFan.associatorOfLimitCone ℬ (tensorObj ℬ W X) Y Z).hom ≫
            (BinaryFan.associatorOfLimitCone ℬ W X (tensorObj ℬ Y Z)).hom) ≫
          (ℬ W (ℬ X (ℬ Y Z).cone.pt).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ X (ℬ Y Z).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ Y Z).cone.π.app { as := WalkingPair.right }

Next tactic:
simp
theorem triangle (X Y : C) :
    (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫
        tensorHom ℬ (𝟙 X) (BinaryFan.leftUnitor 𝒯.isLimit (ℬ 𝒯.cone.pt Y).isLimit).hom =
      tensorHom ℬ (BinaryFan.rightUnitor 𝒯.isLimit (ℬ X 𝒯.cone.pt).isLimit).hom (𝟙 Y) := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X Y : C
⊢ (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫
      tensorHom ℬ (𝟙 X) (BinaryFan.leftUnitor 𝒯.isLimit (ℬ 𝒯.cone.pt Y).isLimit).hom =
    tensorHom ℬ (BinaryFan.rightUnitor 𝒯.isLimit (ℬ X 𝒯.cone.pt).isLimit).hom (𝟙 Y)

Next tactic:
dsimp [tensorHom]
theorem triangle (X Y : C) :
    (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫
        tensorHom ℬ (𝟙 X) (BinaryFan.leftUnitor 𝒯.isLimit (ℬ 𝒯.cone.pt Y).isLimit).hom =
      tensorHom ℬ (BinaryFan.rightUnitor 𝒯.isLimit (ℬ X 𝒯.cone.pt).isLimit).hom (𝟙 Y) := by
  dsimp [tensorHom]
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X Y : C
⊢ (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫
      IsLimit.lift (ℬ X Y).isLimit
        (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ 𝟙 X)
          (BinaryFan.snd (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ BinaryFan.snd (ℬ 𝒯.cone.pt Y).cone)) =
    IsLimit.lift (ℬ X Y).isLimit
      (BinaryFan.mk (BinaryFan.fst (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ BinaryFan.fst (ℬ X 𝒯.cone.pt).cone)
        (BinaryFan.snd (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ 𝟙 Y))

Next tactic:
apply IsLimit.hom_ext (ℬ _ _).isLimit
theorem triangle (X Y : C) :
    (BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫
        tensorHom ℬ (𝟙 X) (BinaryFan.leftUnitor 𝒯.isLimit (ℬ 𝒯.cone.pt Y).isLimit).hom =
      tensorHom ℬ (BinaryFan.rightUnitor 𝒯.isLimit (ℬ X 𝒯.cone.pt).isLimit).hom (𝟙 Y) := by
  dsimp [tensorHom]
  apply IsLimit.hom_ext (ℬ _ _).isLimit; 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : C
𝒯 : LimitCone (Functor.empty C)
ℬ : (X Y : C) → LimitCone (pair X Y)
X Y : C
⊢ ∀ (j : Discrete WalkingPair),
    ((BinaryFan.associatorOfLimitCone ℬ X 𝒯.cone.pt Y).hom ≫
          IsLimit.lift (ℬ X Y).isLimit
            (BinaryFan.mk (BinaryFan.fst (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ 𝟙 X)
              (BinaryFan.snd (ℬ X (ℬ 𝒯.cone.pt Y).cone.pt).cone ≫ BinaryFan.snd (ℬ 𝒯.cone.pt Y).cone))) ≫
        (ℬ X Y).cone.π.app j =
      IsLimit.lift (ℬ X Y).isLimit
          (BinaryFan.mk (BinaryFan.fst (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ BinaryFan.fst (ℬ X 𝒯.cone.pt).cone)
            (BinaryFan.snd (ℬ (ℬ X 𝒯.cone.pt).cone.pt Y).cone ≫ 𝟙 Y)) ≫
        (ℬ X Y).cone.π.app j

Next tactic:
rintro ⟨⟨⟩⟩