Lemma univ_surj_nt_ax : nat_trans_comp _ _ _ p univ_surj_nt = f . Proof. unfold univ_surj_nt; cbn. set (coeq := make_Coequalizer _ _ _ _ _). apply (CoequalizerCommutes coeq). Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.EpiFacts. Require Import UniMath.CategoryTheory.Limits.Coequalizers.