proposition: Lemma is_z_iso_isLim {C : category} {g : graph} (D : diagram g C) (CC : LimCone D) (d : C) (cd : cone D d) : is_z_isomorphism (limArrow CC d cd) -> isLimCone 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 (limArrow CC x cx·iinv). + simpl; intro u. assert (XR:=limArrowCommutes CC x cx u). eapply pathscomp0; [| apply XR]. eapply pathscomp0; [ apply (!assoc _ _ _ ) |]. apply maponpaths. apply z_iso_inv_on_right. apply pathsinv0, limArrowCommutes. - intros p; destruct p as [f Hf]. apply subtypePath. + intro a; apply impred; intro u; apply C. + simpl; apply z_iso_inv_on_left; simpl. apply pathsinv0, limArrowUnique; intro u. cbn in *. eapply pathscomp0; [| apply Hf]. eapply pathscomp0. apply (!assoc _ _ _ ). apply maponpaths. apply limArrowCommutes. 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.Adjunctions.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Limits.Graphs.Colimits