∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y) (n : ℕ),
  ((AlgebraicTopology.alternatingFaceMapComplex C).map f).f n = f.app (Opposite.op (SimplexCategory.mk n))