-- Module: Mathlib.CategoryTheory.Monoidal.PushoutProduct
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasPushouts.{v, u} C
inst_2 : CartesianMonoidalCategory.{v, u} C
inst_3 : MonoidalClosed.{v, u} C
inst_4 : HasInitial.{v, u} C
X : Arrow.{v, u} C
refine_1 :
  CategoryStruct.comp (Y :=
      pushout (X :=
        obj.{v, v, u, u} (obj (curriedTensor C) X.left)
          (Arrow.left.{v, u} (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))))
        (NatTrans.app (map (curriedTensor C) X.hom)
          (Arrow.left.{v, u} (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))))
        (map (obj (curriedTensor C) X.left)
          (Arrow.hom (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))))))
      (pushout.inl
        (NatTrans.app (map (curriedTensor C) X.hom)
          (Arrow.left.{v, u} (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))))
        (map (obj (curriedTensor C) X.left)
          (Arrow.hom (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))))))
      (pushout.desc (IsInitial.to (IsInitial.ofIso initialIsInitial (Iso.symm (zeroMul initialIsInitial))) X.left)
          (Iso.hom (ρ_ X.left)) ⋯ ≫
        CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst) X.left (tensorUnit.{v, u} C (𝒞 := inst)))
          (Iso.inv (ρ_ X.left))
          (pushout.inr
            (NatTrans.app (map (curriedTensor C) X.hom)
              (Arrow.left.{v, u} (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))))
            (map (obj (curriedTensor C) X.left)
              (Arrow.hom (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))))))) =
    pushout.inl
        (NatTrans.app (map (curriedTensor C) X.hom)
          (Arrow.left.{v, u} (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))))
        (map (obj (curriedTensor C) X.left)
          (Arrow.hom (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))))) ≫
      𝟙 (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left
⊢ Eq (α :=
    (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left ⟶
      (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left)
    (pushout.desc (IsInitial.to (IsInitial.ofIso initialIsInitial (Iso.symm (zeroMul initialIsInitial))) X.left)
          (Iso.hom (ρ_ X.left)) ⋯ ≫
        CategoryStruct.comp (Y := tensorObj.{v, u} (𝒞 := inst) X.left (tensorUnit.{v, u} C (𝒞 := inst)))
          (Iso.inv (ρ_ X.left))
          (pushout.inr
            (NatTrans.app (map (curriedTensor C) X.hom)
              (Arrow.left.{v, u} (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))))
            (map (obj (curriedTensor C) X.left)
              (Arrow.hom (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))))) :
      (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left ⟶
        (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left)
    (𝟙 (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left :
      (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left ⟶
        (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left)

Tactic:
refine pushout.hom_ext ?_ (by simp)

Premises used: CategoryTheory.Limits.pushout.hom_ext