Lemma abgr_epi_cokernel_isCokernel_comm {A B C : abgr} (f : abgr_category⟦A, B⟧) (isE : isEpi f) (h : abgr_category⟦A, C⟧) (H : KernelArrow (abgr_Kernel f) · h = ZeroArrow abgr_Zero (abgr_Kernel f) C) : f · abgr_epi_cokernel_out_monoidfun f isE h H = h. Proof. use total2_paths_f. - use funextfun. intros x. use pathsinv0. exact (pr2 (iscontrpr1 (abgr_epi_CokernelOut_iscontr f isE h H (pr1 f x))) (@make_hfiber _ _ (pr1 f) (pr1 f x) x (idpath _))). - use proofirrelevance. use isapropismonoidfun. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Foundations.UnivalenceAxiom. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.Monoids. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.PreAdditive. Require Import UniMath.CategoryTheory.Additive. Require Import UniMath.CategoryTheory.Abelian. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.Kernels. Require Import UniMath.CategoryTheory.Limits.Cokernels. Require Import UniMath.CategoryTheory.Limits.BinDirectSums. @ZeroArrowFrom abgr_category abgr_Zero A = abgrfunfromunit A.