-- 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
W X Y Z : Arrow.{v, u} C
apply :
  Eq (α :=
    obj.{v, v, u, u} (tensorRight Z.right)
        (obj.{v, v, u, u} (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).left) Y.right) ⟶
      (tensorObj (𝒞 := instCategoryArrow) W
          (tensorObj (𝒞 := instCategoryArrow) X (tensorObj (𝒞 := instCategoryArrow) Y Z))).left)
    (CategoryStruct.comp (Y :=
      obj.{v, v, u, u} (tensorRight Z.right)
        (pushout (X := obj.{v, v, u, u} (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).left) Y.left)
          (NatTrans.app (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).hom) Y.left)
          (map (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).left) Y.hom)))
      (map (tensorRight Z.right)
        (pushout.inr (NatTrans.app (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).hom) Y.left)
          (map (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).left) Y.hom)))
      (CategoryStruct.comp (Y :=
        pushout (X :=
          obj.{v, v, u, u}
            (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
            Z.left)
          (NatTrans.app
            (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).hom)
            Z.left)
          (map
            (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
            Z.hom))
        (pushout.inr
          (NatTrans.app
            (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).hom)
            Z.left)
          (map
            (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
            Z.hom))
        (Arrow.Hom.left
          (Iso.hom (α_ W X Y) ▷ Z ≫
            Iso.hom (α_ W (tensorObj (𝒞 := instCategoryArrow) X Y) Z) ≫ W ◁ Iso.hom (α_ X Y Z)))))
    (map (tensorRight Z.right)
        (pushout.inr (NatTrans.app (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).hom) Y.left)
          (map (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) W X).left) Y.hom)) ≫
      pushout.inr
          (NatTrans.app
            (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).hom)
            Z.left)
          (map
            (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
            Z.hom) ≫
        Arrow.Hom.left
          (Iso.hom (α_ (tensorObj (𝒞 := instCategoryArrow) W X) Y Z) ≫
            Iso.hom (α_ W X (tensorObj (𝒞 := instCategoryArrow) Y Z))))
⊢ CategoryStruct.comp (Y :=
      pushout (X :=
        obj.{v, v, u, u}
          (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
          Z.left)
        (NatTrans.app
          (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).hom)
          Z.left)
        (map
          (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
          Z.hom))
      (pushout.inr
        (NatTrans.app
          (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).hom)
          Z.left)
        (map
          (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
          Z.hom))
      (Arrow.Hom.left
        (Iso.hom (α_ W X Y) ▷ Z ≫ Iso.hom (α_ W (tensorObj (𝒞 := instCategoryArrow) X Y) Z) ≫ W ◁ Iso.hom (α_ X Y Z))) =
    pushout.inr
        (NatTrans.app
          (map (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).hom)
          Z.left)
        (map
          (obj (curriedTensor C) (tensorObj (𝒞 := instCategoryArrow) (tensorObj (𝒞 := instCategoryArrow) W X) Y).left)
          Z.hom) ≫
      Arrow.Hom.left
        (Iso.hom (α_ (tensorObj (𝒞 := instCategoryArrow) W X) Y Z) ≫
          Iso.hom (α_ W X (tensorObj (𝒞 := instCategoryArrow) Y Z)))

Tactic:
apply ((tensorRight _).map_isPushout (IsPushout.of_hasPushout _ _)).hom_ext (by simp)

Premises used: CategoryTheory.IsPushout.of_hasPushout, CategoryTheory.MonoidalCategory.tensorRight, CategoryTheory.IsPushout.hom_ext, CategoryTheory.Functor.map_isPushout