Definition AddEquivCounitInvMor_is_iso_with_inv_data {A1 A2 : CategoryWithAdditiveStructure} (AE : AddEquiv A1 A2) (X : A2) : is_z_isomorphism (counit_from_left_adjoint AE X) := ((dirprod_pr2 (pr2 AE)) X).
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.