-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal (goal)
Goal:
X : SSet
sx : X.StrictSegal
n : ℕ
s :
  Cone (J := StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
    (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X)
x : s.pt
i : ℕ
hi : i ≤ 2
f :
  unop
      (obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.op (inclusion 2))
        (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋i⦌ hi))) ⟶
    unop (op ⦋n⦌)
k : Fin i
⊢ Eq (α := X _⦋1⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (mkOfSucc k).op) : (a : X _⦋i⦌) → X _⦋1⦌)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (op f)) : (a : X _⦋n⦌) → X _⦋i⦌)
            (sx.spineToSimplex
              {
                vertex := fun (i : Fin (n + 1)) ↦
                  (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                        (s.π.app
                          (StructuredArrow.mk (Y :=
                            op
                              (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                                lift._proof_2))
                            (Quiver.Hom.op (⦋0⦌.const ⦋n⦌ i)))) :
                      (a : s.pt) → X _⦋0⦌)
                    x,
                arrow := fun (i : Fin n) ↦
                  (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                        (s.π.app
                          (StructuredArrow.mk (Y :=
                            op
                              (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌
                                lift._proof_4))
                            (mkOfLe i.castSucc i.succ ⋯).op)) :
                      (a : s.pt) → X _⦋1⦌)
                    x,
                arrow_src := ⋯, arrow_tgt := ⋯ }) :
          X _⦋i⦌) :
      X _⦋1⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (mkOfSucc k).op) : (a : X _⦋i⦌) → X _⦋1⦌)
        (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (s.π.app (StructuredArrow.mk (op f))) :
                TypeCat.Fun s.pt (X _⦋i⦌)) :
              (a : s.pt) → X _⦋i⦌)
            x :
          X _⦋i⦌) :
      X _⦋1⦌)
have h : Eq (α := X _⦋1⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (op f ≫ (mkOfSucc k).op)) : (a : X _⦋n⦌) → X _⦋1⦌)
      (sx.spineToSimplex
        {
          vertex := fun (i : Fin (n + 1)) ↦
            (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                  (s.π.app
                    (StructuredArrow.mk (Y :=
                      op
                        (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌
                          lift._proof_2))
                      (Quiver.Hom.op (⦋0⦌.const ⦋n⦌ i)))) :
                (a : s.pt) → X _⦋0⦌)
              x,
          arrow := fun (i : Fin n) ↦
            (@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                  (s.π.app
                    (StructuredArrow.mk (Y :=
                      op
                        (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌
                          lift._proof_4))
                      (mkOfLe i.castSucc i.succ sorry).op)) :
                (a : s.pt) → X _⦋1⦌)
              x,
          arrow_src := sorry, arrow_tgt := sorry }) :
    X _⦋1⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (mkOfSucc k).op) : (a : X _⦋i⦌) → X _⦋1⦌)
      (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _ (s.π.app (StructuredArrow.mk (op f))) :
              TypeCat.Fun s.pt (X _⦋i⦌)) :
            (a : s.pt) → X _⦋i⦌)
          x :
        X _⦋i⦌) :
    X _⦋1⦌)
Tactic: rw [← Functor.map_comp_apply]