-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.PushoutProduct (goal)
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
⊢ Iso.hom
        {
          hom :=
            pushout.desc (IsInitial.to (IsInitial.ofIso initialIsInitial (Iso.symm (zeroMul initialIsInitial))) X.left)
              (Iso.hom (ρ_ X.left)) ⋯,
          inv :=
            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))))))),
          hom_inv_id := ⋯, inv_hom_id := ⋯ } ≫
      X.hom =
    (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).hom ≫ Iso.hom (ρ_ X.right)
have refine_2 : 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)))))) :
      obj.{v, v, u, u} (obj (curriedTensor C) X.right)
          (Arrow.left.{v, u} (Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst))))) ⟶
        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)))))))
    (Iso.hom
          {
            hom :=
              pushout.desc
                (IsInitial.to (IsInitial.ofIso initialIsInitial (Iso.symm (zeroMul initialIsInitial))) X.left)
                (Iso.hom (ρ_ X.left)) sorry,
            inv :=
              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))))))),
            hom_inv_id := sorry, inv_hom_id := sorry } ≫
        X.hom :
      (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).left ⟶ X.right) =
  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)))))) ≫
    (X □ Arrow.mk (X := initial.{v, u} C) (initial.to (tensorUnit.{v, u} C (𝒞 := inst)))).hom ≫ Iso.hom (ρ_ X.right)
Tactic: refine pushout.hom_ext ?_ (by simp)