Lemma S_imp_SS h n : # L inF · h = ψ μF h → # L (e n) · h = Pow n IC z. Proof. intros Hh. induction n. - cbn. apply (InitialArrowUnique ILD). - rewrite e_comm, functor_comp, <- assoc, Hh. assert (H : # L (# F (e n)) · ψ μF h = ψ (iter_functor F n 0) (# L (e n) · h)). { apply pathsinv0, (eqtohomot (nat_trans_ax ψ _ _ (e n))). } now rewrite H, IHn. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Chains.Chains. Require Import UniMath.CategoryTheory.Chains.Adamek. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.whiskering.