Lemma isColim_is_z_iso {C : precategory} {g : graph} (D : diagram g C) (CC : ColimCocone D) (d : C) (cd : cocone D d) : isColimCocone D d cd -> is_z_isomorphism (colimArrow CC d cd). Proof. intro H. set (CD := make_ColimCocone D d cd H). apply (tpair _ (colimArrow (make_ColimCocone D d cd H) (colim CC) (colimCocone CC))). abstract (split; [ apply pathsinv0, colim_endo_is_identity; simpl; intro u; rewrite assoc; eapply pathscomp0; [eapply cancel_postcomposition; apply colimArrowCommutes|]; apply (colimArrowCommutes CD) | apply pathsinv0, (colim_endo_is_identity _ CD); simpl; intro u; rewrite assoc; eapply pathscomp0; [eapply cancel_postcomposition; apply (colimArrowCommutes CD)|]; apply colimArrowCommutes ]). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Adjunctions.Core.