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

Tactic:
obtain ⟨y, rfl⟩ := y.mk_surjective

Premises used: SSet.Truncated.HomotopyCategory.mk_surjective, rfl