proposition: Lemma isaprop_Colims: isaprop (Colims C).
proof: 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.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Propositions UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Adjunctions.Core