proposition: Lemma isLim_is_z_iso {g : graph} (D : diagram g C^op) (CC : LimCone D) (d : C) (cd : cone D d) : isLimCone D d cd -> is_z_isomorphism (limArrow CC d cd).
proof: Proof. intro H. set (CD := make_LimCone D d cd H). apply (tpair _ (limArrow (make_LimCone D d cd H) (lim CC) (limCone CC))). split. apply pathsinv0. change d with (lim CD). apply lim_endo_is_identity. simpl; intro u; rewrite <- assoc. eapply pathscomp0; [eapply maponpaths; apply limArrowCommutes|]. apply (limArrowCommutes CC). apply pathsinv0, (lim_endo_is_identity _ CC); simpl; intro u; rewrite <- assoc. eapply pathscomp0; [eapply maponpaths; apply (limArrowCommutes CC)|]. apply (limArrowCommutes CD). 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