proposition: Lemma equiv_isCoequalizer1 {a b : C} {f g : C⟦a, b⟧} (e : C) (h : C⟦b, e⟧) (H : f · h = g · h) : Limits.Coequalizers.isCoequalizer f g h H -> isCoequalizer C f g e h H.
proof: Proof. intros X. set (E := Limits.Coequalizers.make_Coequalizer f g h H X). use (make_isCoequalizer C). intros e' h' H'. use (unique_exists (Limits.Coequalizers.CoequalizerOut E e' h' H')). - exact (Limits.Coequalizers.CoequalizerCommutes E e' h' H'). - intros y. apply C. - intros y T. cbn in T. use (Limits.Coequalizers.CoequalizerOutsEq E). use (pathscomp0 T). exact (!(Limits.Coequalizers.CoequalizerCommutes E e' h' H')). Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.Combinatorics.StandardFiniteSets UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.EqDiag UniMath.CategoryTheory.Limits.Coequalizers