Lemma free_ptd_alg_counit_forms_cocone (X : ptd_alg F) : forms_cocone (iter_chain_at X) (free_ptd_alg_counit_cocone_arrows X). Proof. red; intros i _ []; simpl. set (ɛ := free_ptd_alg_counit_cocone_arrows). rewrite assoc. induction i. - simpl. rewrite functor_id, id_right. apply ptd_alg_law. - simpl. rewrite <- functor_comp, assoc. apply (maponpaths (λ x, # F x · _) IHi). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.Chains. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering.