-- Module: Mathlib.CategoryTheory.Monoidal.Arrow
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasPushouts.{v, u} C
inst_2 : HasInitial.{v, u} C
inst_3 : CartesianMonoidalCategory.{v, u} C
inst_4 : MonoidalClosed.{v, u} C
inst_5 : BraidedCategory.{v, u} C
X₁ X₂ X₃ Y₁ Y₂ Y₃ : Arrow.{v, u} C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
h₁ :
  Eq (α :=
    tensorObj.{v, u} (𝒞 := inst) (tensorObj.{v, u} (𝒞 := inst) X₁.left X₂.right) X₃.right ⟶
      tensorObj.{v, u} (𝒞 := inst) Y₁.left (tensorObj.{v, u} (𝒞 := inst) Y₂.right Y₃.right))
    (Iso.hom (α_ X₁.left X₂.right X₃.right) ≫
      X₁.left ◁ X₂.right ◁ Arrow.Hom.right f₃ ≫
        Iso.inv (α_ X₁.left X₂.right Y₃.right) ≫
          Arrow.Hom.left f₁ ▷ X₂.right ▷ Y₃.right ≫
            Iso.hom (α_ Y₁.left X₂.right Y₃.right) ≫ Y₁.left ◁ Arrow.Hom.right f₂ ▷ Y₃.right)
    (Arrow.Hom.left f₁ ▷ X₂.right ▷ X₃.right ≫
      Iso.hom (α_ Y₁.left X₂.right X₃.right) ≫
        Y₁.left ◁ Arrow.Hom.right f₂ ▷ X₃.right ≫ Y₁.left ◁ Y₂.right ◁ Arrow.Hom.right f₃)
⊢ Eq (α :=
    obj.{v, v, u, u} (tensorRight X₃.right) (obj.{v, v, u, u} (obj (curriedTensor C) X₁.left) X₂.right) ⟶
      (tensorObj (𝒞 := instCategoryArrow) Y₁ (tensorObj (𝒞 := instCategoryArrow) Y₂ Y₃)).left)
    (CategoryStruct.comp (Y :=
      obj.{v, v, u, u} (tensorRight X₃.right)
        (pushout (X := obj.{v, v, u, u} (obj (curriedTensor C) X₁.left) X₂.left)
          (NatTrans.app (map (curriedTensor C) X₁.hom) X₂.left) (map (obj (curriedTensor C) X₁.left) X₂.hom)))
      (map (tensorRight X₃.right)
        (pushout.inr (NatTrans.app (map (curriedTensor C) X₁.hom) X₂.left)
          (map (obj (curriedTensor C) X₁.left) X₂.hom)))
      (CategoryStruct.comp (Y :=
        pushout (X := obj.{v, v, u, u} (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).left) X₃.left)
          (NatTrans.app (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).hom) X₃.left)
          (map (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).left) X₃.hom))
        (pushout.inr (NatTrans.app (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).hom) X₃.left)
          (map (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).left) X₃.hom))
        (Arrow.Hom.left (((f₁ ⊗ₘ f₂) ⊗ₘ f₃) ≫ Iso.hom (α_ Y₁ Y₂ Y₃)))))
    (CategoryStruct.comp (Y :=
      obj.{v, v, u, u} (tensorRight X₃.right)
        (pushout (X := obj.{v, v, u, u} (obj (curriedTensor C) X₁.left) X₂.left)
          (NatTrans.app (map (curriedTensor C) X₁.hom) X₂.left) (map (obj (curriedTensor C) X₁.left) X₂.hom)))
      (map (tensorRight X₃.right)
        (pushout.inr (NatTrans.app (map (curriedTensor C) X₁.hom) X₂.left)
          (map (obj (curriedTensor C) X₁.left) X₂.hom)))
      (CategoryStruct.comp (Y :=
        pushout (X := obj.{v, v, u, u} (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).left) X₃.left)
          (NatTrans.app (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).hom) X₃.left)
          (map (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).left) X₃.hom))
        (pushout.inr (NatTrans.app (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).hom) X₃.left)
          (map (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) X₁ X₂).left) X₃.hom))
        (Arrow.Hom.left (Iso.hom (α_ X₁ X₂ X₃) ≫ (f₁ ⊗ₘ f₂ ⊗ₘ f₃)))))

Tactic:
suffices ((α_ _ _ _).hom ≫ _ ◁ _ ◁ f₃.right ≫ (α_ _ _ _).inv ≫ f₁.left ▷ _ ▷ _ ≫
      (α_ _ _ _).hom ≫ _ ◁ f₂.right ▷ _ = f₁.left ▷ _ ▷ _ ≫ (α_ _ _ _).hom ≫
      _ ◁ f₂.right ▷ _ ≫ _ ◁ _ ◁ f₃.right) by
      simp [← whisker_exchange_assoc, reassoc_of% this]