-- Module: Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal
Goal:
X X' Y Y' Z : Truncated 2
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
z : Z.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.curry₃
                      (Equivalence.inverse
                          (prod.associativity 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 x₀)).obj
              (mk y)).obj
          z ⟶
        (((Functor.obj Functor.curry₃
                      (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                        inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                  (mk x₁)).obj
              (mk y)).obj
          z)
      (NatTrans.app
        (NatTrans.app
          ((Functor.obj Functor.curry₃
                  (Equivalence.inverse (prod.associativity 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)))).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))) ↦
                        mkNatIso
                          (fun
                              (z :
                                Z.obj
                                  (Opposite.op
                                    (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                                      OneTruncation₂._proof_1))) ↦
                            Iso.refl
                              ((((Functor.obj Functor.curry₃
                                            (Equivalence.inverse
                                                (prod.associativity 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 z)))
                          ⋯)
                      ⋯ :
                    (Functor.obj Functor.curry₃
                            (Equivalence.inverse
                                (prod.associativity 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) ≅
                      (Functor.obj Functor.curry₃
                            (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                              inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                        (mk x)))
                x₁))
          (mk y))
        z)
      (NatTrans.app
        (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))) ↦
                        mkNatIso
                          (fun
                              (z :
                                Z.obj
                                  (Opposite.op
                                    (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                                      OneTruncation₂._proof_1))) ↦
                            Iso.refl
                              ((((Functor.obj Functor.curry₃
                                            (Equivalence.inverse
                                                (prod.associativity 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 z)))
                          ⋯)
                      ⋯ :
                    (Functor.obj Functor.curry₃
                            (Equivalence.inverse
                                (prod.associativity 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) ≅
                      (Functor.obj Functor.curry₃
                            (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                              inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                        (mk x)))
                x₀) ≫
            (Functor.obj Functor.curry₃
                  (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                    inverse X (tensorObj (𝒞 := Functor.category) Y Z))).map
              (homMk e))
          (mk y))
        z)
⊢ Eq (α :=
    (((Functor.obj Functor.curry₃
                    (Equivalence.inverse (prod.associativity 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 x₀)).obj
            y).obj
        z ⟶
      (((Functor.obj Functor.curry₃
                    (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                      inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                (mk x₁)).obj
            y).obj
        z)
    (NatTrans.app
      (NatTrans.app
        ((Functor.obj Functor.curry₃
                (Equivalence.inverse (prod.associativity 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)))).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))) ↦
                      mkNatIso
                        (fun
                            (z :
                              Z.obj
                                (Opposite.op
                                  (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                                    OneTruncation₂._proof_1))) ↦
                          Iso.refl
                            ((((Functor.obj Functor.curry₃
                                          (Equivalence.inverse
                                              (prod.associativity 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 z)))
                        ⋯)
                    ⋯ :
                  (Functor.obj Functor.curry₃
                          (Equivalence.inverse
                              (prod.associativity 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) ≅
                    (Functor.obj Functor.curry₃
                          (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                            inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                      (mk x)))
              x₁))
        y)
      z)
    (NatTrans.app
      (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))) ↦
                      mkNatIso
                        (fun
                            (z :
                              Z.obj
                                (Opposite.op
                                  (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                                    OneTruncation₂._proof_1))) ↦
                          Iso.refl
                            ((((Functor.obj Functor.curry₃
                                          (Equivalence.inverse
                                              (prod.associativity 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 z)))
                        ⋯)
                    ⋯ :
                  (Functor.obj Functor.curry₃
                          (Equivalence.inverse
                              (prod.associativity 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) ≅
                    (Functor.obj Functor.curry₃
                          (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                            inverse X (tensorObj (𝒞 := Functor.category) Y Z))).obj
                      (mk x)))
              x₀) ≫
          (Functor.obj Functor.curry₃
                (Functor.prod (𝟭 X.HomotopyCategory) (inverse Y Z) ⋙
                  inverse X (tensorObj (𝒞 := Functor.category) Y Z))).map
            (homMk e))
        y)
      z)

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

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