Lemma isaprop_Colims: isaprop (Colims C). Proof. apply impred; intro g; apply impred; intro cc. apply invproofirrelevance; intros Hccx Hccy. apply subtypePath. - intro; apply isaprop_isColimCocone. - apply (total2_paths_f (isotoid _ H (z_iso_from_colim_to_colim Hccx Hccy))). set (B c := ∏ v, C⟦dob cc v,c⟧). set (C' (c : C) f := forms_cocone(c:=c) cc f). rewrite (@transportf_total2 _ B C'). apply subtypePath. + intro; repeat (apply impred; intro); apply univalent_category_has_homsets. + simpl; eapply pathscomp0; [apply transportf_isotoid_dep''|]. apply funextsec; intro v. now rewrite idtoiso_isotoid; apply colimArrowCommutes. Qed.
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.