-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal (goal)
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
⊢ (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)
have 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)))
                  sorry :
                (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)))
                  sorry :
                (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
Tactic: ext y