proposition: Lemma equiv_isEqualizer1 {a b : C} {f g : C⟦a, b⟧} (e : C) (h : C⟦e, a⟧) (H : h · f = h · g) : Limits.Equalizers.isEqualizer f g h H -> isEqualizer C f g e h H.
proof: Proof. intros X. set (E := Limits.Equalizers.make_Equalizer f g h H X). use (make_isEqualizer C). intros e' h' H'. use (unique_exists (Limits.Equalizers.EqualizerIn E e' h' H')). - exact (Limits.Equalizers.EqualizerCommutes E e' h' H'). - intros y. apply hs. - intros y T. cbn in T. use (Limits.Equalizers.EqualizerInsEq E). use (pathscomp0 T). exact (!(Limits.Equalizers.EqualizerCommutes 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.Isos UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Equalizers