∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] (X : CategoryTheory.Functor AugmentedSimplexCategory C),
  (AugmentedSimplexCategory.equivAugmentedCosimplicialObjectFunctorCompToArrowIso.inv.app X).left =
    CategoryTheory.CategoryStruct.id (X.obj CategoryTheory.WithInitial.star)