-- Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
theorem associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) :
    tensorHom ℬ (tensorHom ℬ f₁ f₂) f₃ ≫ (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom =
      (BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫ tensorHom ℬ f₁ (tensorHom ℬ f₂ f₃)

Proof state:
case mk.right.mk.left
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ (((ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift
            (BinaryFan.mk
              (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫
                (ℬ Y₁ Y₂).isLimit.lift
                  (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)))
              (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫
          (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom) ≫
        (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ Y₂ Y₃).cone.π.app { as := WalkingPair.left } =
    (((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
          (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift
            (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁)
              (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫
                (ℬ Y₂ Y₃).isLimit.lift
                  (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))) ≫
        (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
      (ℬ Y₂ Y₃).cone.π.app { as := WalkingPair.left }

Tactic:
simp

Result:
no goals

Proof state:
case mk.right
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ ((ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift
          (BinaryFan.mk
            (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫
              (ℬ Y₁ Y₂).isLimit.lift
                (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)))
            (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫
        (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom) ≫
      (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.right } =
    ((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
        (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift
          (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁)
            (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫
              (ℬ Y₂ Y₃).isLimit.lift
                (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))) ≫
      (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.right }

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

Result:
case mk.right
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ ∀ (j : Discrete WalkingPair),
    (((ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift
              (BinaryFan.mk
                (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫
                  (ℬ Y₁ Y₂).isLimit.lift
                    (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)))
                (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫
            (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom) ≫
          (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ Y₂ Y₃).cone.π.app j =
      (((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
            (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift
              (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁)
                (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫
                  (ℬ Y₂ Y₃).isLimit.lift
                    (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))) ≫
          (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.right }) ≫
        (ℬ Y₂ Y₃).cone.π.app j

Proof state:
case mk.left
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ ((ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift
          (BinaryFan.mk
            (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫
              (ℬ Y₁ Y₂).isLimit.lift
                (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)))
            (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫
        (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom) ≫
      (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.left } =
    ((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
        (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift
          (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁)
            (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫
              (ℬ Y₂ Y₃).isLimit.lift
                (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))) ≫
      (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app { as := WalkingPair.left }

Tactic:
simp

Result:
no goals

Proof state:
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ (ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift
        (BinaryFan.mk
          (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫
            (ℬ Y₁ Y₂).isLimit.lift
              (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)))
          (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫
      (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom =
    (BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
      (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift
        (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁)
          (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫
            (ℬ Y₂ Y₃).isLimit.lift
              (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))

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

Result:
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X₁ X₂ X₃ Y₁ Y₂ Y₃ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
⊢ ∀ (j : Discrete WalkingPair),
    ((ℬ (tensorObj ℬ Y₁ Y₂) Y₃).isLimit.lift
            (BinaryFan.mk
              (BinaryFan.fst (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫
                (ℬ Y₁ Y₂).isLimit.lift
                  (BinaryFan.mk (BinaryFan.fst (ℬ X₁ X₂).cone ≫ f₁) (BinaryFan.snd (ℬ X₁ X₂).cone ≫ f₂)))
              (BinaryFan.snd (ℬ (tensorObj ℬ X₁ X₂) X₃).cone ≫ f₃)) ≫
          (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom) ≫
        (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app j =
      ((BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫
          (ℬ Y₁ (tensorObj ℬ Y₂ Y₃)).isLimit.lift
            (BinaryFan.mk (BinaryFan.fst (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫ f₁)
              (BinaryFan.snd (ℬ X₁ (ℬ X₂ X₃).cone.pt).cone ≫
                (ℬ Y₂ Y₃).isLimit.lift
                  (BinaryFan.mk (BinaryFan.fst (ℬ X₂ X₃).cone ≫ f₂) (BinaryFan.snd (ℬ X₂ X₃).cone ≫ f₃))))) ≫
        (ℬ Y₁ (ℬ Y₂ Y₃).cone.pt).cone.π.app j