-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal (goal)
Goal:
X X' Y Y' Z : Truncated 2
x :
  X.obj
    (Opposite.op
      (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌ OneTruncation₂._proof_1))
y :
  Y.obj
    (Opposite.op
      (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌ OneTruncation₂._proof_1))
z₀ z₁ :
  Z.obj
    (Opposite.op
      (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌ OneTruncation₂._proof_1))
e : Edge z₀ z₁
⊢ Eq (α :=
    (((Functor.obj Functor.curry₃
                    (prod.inverseAssociator X.HomotopyCategory Y.HomotopyCategory Z.HomotopyCategory ⋙
                      (inverse X Y).prod (𝟭 Z.HomotopyCategory) ⋙
                        inverse (tensorObj (𝒞 := Functor.category) X Y) Z ⋙
                          mapHomotopyCategory (Iso.hom (α_ X Y Z)))).obj
                (mk x)).obj
            (mk y)).obj
        (mk z₀) ⟶
      (((Functor.obj Functor.curry₃
                    (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                      inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                (mk x)).obj
            (mk y)).obj
        (mk z₁))
    (CategoryTheory.Functor.map (mapHomotopyCategory (Iso.hom (α_ X Y Z)))
          (CategoryTheory.Functor.map (inverse (tensorObj (𝒞 := Functor.category) X Y) Z)
            (Prod.mkHom ((inverse X Y).map (Prod.mkHom (𝟙 (mk x)) (𝟙 (mk y)))) (homMk e))) ≫
        𝟙
          ((((Functor.obj Functor.curry₃
                        (prod.inverseAssociator X.HomotopyCategory Y.HomotopyCategory Z.HomotopyCategory ⋙
                          (inverse X Y).prod (𝟭 Z.HomotopyCategory) ⋙
                            inverse (tensorObj (𝒞 := Functor.category) X Y) Z ⋙
                              mapHomotopyCategory (Iso.hom (α_ X Y Z)))).obj
                    (mk x)).obj
                (mk y)).obj
            (mk z₁)) :
      (((Functor.obj Functor.curry₃
                      (prod.inverseAssociator X.HomotopyCategory Y.HomotopyCategory Z.HomotopyCategory ⋙
                        (inverse X Y).prod (𝟭 Z.HomotopyCategory) ⋙
                          inverse (tensorObj (𝒞 := Functor.category) X Y) Z ⋙
                            mapHomotopyCategory (Iso.hom (α_ X Y Z)))).obj
                  (mk x)).obj
              (mk y)).obj
          (mk z₀) ⟶
        (((Functor.obj Functor.curry₃
                      (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                        inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                  (mk x)).obj
              (mk y)).obj
          (mk z₁))
    (𝟙
          ((((Functor.obj Functor.curry₃
                        (prod.inverseAssociator X.HomotopyCategory Y.HomotopyCategory Z.HomotopyCategory ⋙
                          (inverse X Y).prod (𝟭 Z.HomotopyCategory) ⋙
                            inverse (tensorObj (𝒞 := Functor.category) X Y) Z ⋙
                              mapHomotopyCategory (Iso.hom (α_ X Y Z)))).obj
                    (mk x)).obj
                (mk y)).obj
            (mk z₀)) ≫
        CategoryTheory.Functor.map (inverse X (tensorObj (𝒞 := Functor.category) Y Z))
          (Prod.mkHom (𝟙 (mk x)) ((inverse Y Z).map (𝟙 (mk y), homMk e))) :
      (((Functor.obj Functor.curry₃
                      (prod.inverseAssociator X.HomotopyCategory Y.HomotopyCategory Z.HomotopyCategory ⋙
                        (inverse X Y).prod (𝟭 Z.HomotopyCategory) ⋙
                          inverse (tensorObj (𝒞 := Functor.category) X Y) Z ⋙
                            mapHomotopyCategory (Iso.hom (α_ X Y Z)))).obj
                  (mk x)).obj
              (mk y)).obj
          (mk z₀) ⟶
        (((Functor.obj Functor.curry₃
                      (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                        inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                  (mk x)).obj
              (mk y)).obj
          (mk z₁))
have rw : CategoryTheory.Functor.map (mapHomotopyCategory (Iso.hom (α_ X Y Z)))
    (CategoryTheory.Functor.map (inverse (tensorObj (𝒞 := Functor.category) X Y) Z)
      (Prod.mkHom (𝟙 ((inverse X Y).obj (mk x, mk y))) (homMk e))) =
  homMk ((Edge.id x).tensor ((Edge.id y).tensor e))
Tactic: rw [Category.comp_id, Category.id_comp, ← prod_id,
          inverse_map_mkHom_id_homMk, inverse_map_mkHom_id_homMk,
          CategoryTheory.Functor.map_id]