proposition: Lemma is_z_iso_isColim {C : category} {g : graph} (D : diagram g C) (CC : ColimCocone D) (d : C) (cd : cocone D d) : is_z_isomorphism (colimArrow CC d cd) -> isColimCocone D d cd.
proof: Proof. intro H. set (iinv := z_iso_inv_from_is_z_iso _ H). intros x cx. use tpair. - use tpair. + exact (iinv · colimArrow CC x cx). + simpl; intro u. rewrite <- (colimArrowCommutes CC x cx u), assoc. apply cancel_postcomposition, pathsinv0, z_iso_inv_on_left, pathsinv0, colimArrowCommutes. - intros p; destruct p as [f Hf]. apply subtypePath. + intro a; apply impred; intro u; apply C. + simpl; apply pathsinv0, z_iso_inv_on_right; simpl. apply pathsinv0, colimArrowUnique; intro u. now rewrite <- (Hf u), assoc, colimArrowCommutes. Defined.
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