Lemma chain_mor_coconeIn {C : category} (c : chain C) (x : C) (cc : cocone c x) i : ∏ j (Hij : i < j), chain_mor c Hij · coconeIn cc j = coconeIn cc i. Proof. induction j as [|j IHj]. - intros Hi0; destruct (negnatlthn0 _ Hi0). - intros Hij; simpl. destruct (natlehchoice4 _ _ Hij). + rewrite <- (IHj h), <- assoc. apply maponpaths, coconeInCommutes. + destruct p. apply coconeInCommutes. Qed.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.Graphs.Limits. Require Import UniMath.CategoryTheory.Limits.Initial.