-- Module: Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal
Goal:
X X' Y Y' Z : Truncated 2
f : X ⟶ X'
x₀ x₁ :
  X.obj
    (Opposite.op
      (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌ OneTruncation₂._proof_1))
e : Edge x₀ x₁
w :
  ∀ (y : Y.HomotopyCategory),
    NatTrans.app
        ((Functor.obj Functor.curry ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).map
            (homMk e) ≫
          Iso.hom
            ((fun
                  (x :
                    X.obj
                      (Opposite.op
                        (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                          OneTruncation₂._proof_1))) ↦
                (mkNatIso
                    (fun
                        (y :
                          Y.obj
                            (Opposite.op
                              (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                                OneTruncation₂._proof_1))) ↦
                      Iso.refl
                        (((Functor.obj Functor.curry
                                  ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                              (mk x)).obj
                          (mk y)))
                    ⋯ :
                  (Functor.obj Functor.curry ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                      (mk x) ≅
                    (Functor.obj Functor.curry (inverse X Y ⋙ mapHomotopyCategory (f ▷ Y))).obj (mk x)))
              x₁))
        y =
      NatTrans.app
        (Iso.hom
            ((fun
                  (x :
                    X.obj
                      (Opposite.op
                        (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                          OneTruncation₂._proof_1))) ↦
                (mkNatIso
                    (fun
                        (y :
                          Y.obj
                            (Opposite.op
                              (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                                OneTruncation₂._proof_1))) ↦
                      Iso.refl
                        (((Functor.obj Functor.curry
                                  ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                              (mk x)).obj
                          (mk y)))
                    ⋯ :
                  (Functor.obj Functor.curry ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                      (mk x) ≅
                    (Functor.obj Functor.curry (inverse X Y ⋙ mapHomotopyCategory (f ▷ Y))).obj (mk x)))
              x₀) ≫
          (Functor.obj Functor.curry (inverse X Y ⋙ mapHomotopyCategory (f ▷ Y))).map (homMk e))
        y
⊢ (Functor.obj Functor.curry ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).map (homMk e) ≫
      Iso.hom
        ((fun
              (x :
                X.obj
                  (Opposite.op
                    (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                      OneTruncation₂._proof_1))) ↦
            (mkNatIso
                (fun
                    (y :
                      Y.obj
                        (Opposite.op
                          (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                            OneTruncation₂._proof_1))) ↦
                  Iso.refl
                    (((Functor.obj Functor.curry
                              ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                          (mk x)).obj
                      (mk y)))
                ⋯ :
              (Functor.obj Functor.curry ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                  (mk x) ≅
                (Functor.obj Functor.curry (inverse X Y ⋙ mapHomotopyCategory (f ▷ Y))).obj (mk x)))
          x₁) =
    Iso.hom
        ((fun
              (x :
                X.obj
                  (Opposite.op
                    (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                      OneTruncation₂._proof_1))) ↦
            (mkNatIso
                (fun
                    (y :
                      Y.obj
                        (Opposite.op
                          (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                            OneTruncation₂._proof_1))) ↦
                  Iso.refl
                    (((Functor.obj Functor.curry
                              ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                          (mk x)).obj
                      (mk y)))
                ⋯ :
              (Functor.obj Functor.curry ((mapHomotopyCategory f).prod (𝟭 Y.HomotopyCategory) ⋙ inverse X' Y)).obj
                  (mk x) ≅
                (Functor.obj Functor.curry (inverse X Y ⋙ mapHomotopyCategory (f ▷ Y))).obj (mk x)))
          x₀) ≫
      (Functor.obj Functor.curry (inverse X Y ⋙ mapHomotopyCategory (f ▷ Y))).map (homMk e)

Tactic:
ext y