-- Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
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)

Proof state:
C : Type u
inst✝ : Category.{v, u} 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 ≫
      (ℬ X Y).isLimit.lift
        (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).isLimit.lift
      (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))

Tactic:
apply IsLimit.hom_ext (ℬ _ _).isLimit

Result:
C : Type u
inst✝ : Category.{v, u} 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 ≫
          (ℬ X Y).isLimit.lift
            (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 =
      (ℬ X Y).isLimit.lift
          (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

Proof state:
C : Type u
inst✝ : Category.{v, u} 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 ≫
          (ℬ X Y).isLimit.lift
            (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 =
      (ℬ X Y).isLimit.lift
          (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

Tactic:
rintro ⟨⟨⟩⟩ <;> simp

Result:
no goals

Proof state:
C : Type u
inst✝ : Category.{v, u} 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)

Tactic:
dsimp [tensorHom]

Result:
C : Type u
inst✝ : Category.{v, u} 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 ≫
      (ℬ X Y).isLimit.lift
        (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).isLimit.lift
      (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))