Lemma AddEquivCounitMorComm {A1 A2 : CategoryWithAdditiveStructure} (AE : AddEquiv A1 A2) {x x' : ob A2} (f : x --> x') : f = (inv_from_z_iso (AddEquivCounitIso AE x)) · (# (functor_composite (AddEquiv2 AE) (AddEquiv1 AE)) f) · (AddEquivCounitIso AE x'). Proof. use (pre_comp_with_z_iso_is_inj (AddEquivCounitIso AE x)). use (pathscomp0 (! AddEquivCounitComm AE _ _ f)). rewrite assoc. rewrite assoc. set (tmp := is_inverse_in_precat1 (AddEquivCounitIso AE x)). rewrite tmp. rewrite id_left. apply idpath. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.CategoriesWithBinOps. Require Import UniMath.CategoryTheory.PrecategoriesWithAbgrops. Require Import UniMath.CategoryTheory.PreAdditive. Require Import UniMath.CategoryTheory.Additive. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Limits.Zero. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.BinDirectSums. exists (ZeroArrowFrom (F (to_Zero A))). apply ArrowsFromZero. assert (e2 : # F (ZeroArrowFrom a2) = @ZeroArrowFrom B Z (F a2)). apply (ArrowsFromZero B Z). use (FromBinDirectSumsEq B BDS); cbn.