proposition: Lemma slice_precat_isColimCocone (d : diagram g (C / x)) (a : C / x) (cc : cocone d a) (H : isColimCocone (mapdiagram U d) (U a) (mapcocone U d cc)) : isColimCocone d a cc.
proof: Proof. set (CC := make_ColimCocone _ _ _ H). intros y ccy. use unique_exists. + use tpair; simpl. * apply (colimArrow CC), (mapcocone U _ ccy). * abstract (apply pathsinv0; eapply pathscomp0; [apply (postcompWithColimArrow _ CC (pr1 y) (mapcocone U d ccy))|]; apply pathsinv0, (colimArrowUnique CC); intros u; simpl; eapply pathscomp0; [apply (!(pr2 (coconeIn cc u)))|]; apply (pr2 (coconeIn ccy u))). + abstract (intros u; apply subtypePath; [intros xx; apply C|]; simpl; apply (colimArrowCommutes CC)). + abstract (intros f; simpl; apply impred; intro u; apply has_homsets_slice_precat). + abstract (intros f; simpl; intros Hf; apply eq_mor_slicecat; simpl; apply (colimArrowUnique CC); intro u; cbn; rewrite <- (Hf u); apply idpath). Defined.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Univalence UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis