proposition: Lemma AddEquivCounitUnit' {A1 A2 : CategoryWithAdditiveStructure} (AE : AddEquiv A1 A2) (x : A1) : ((AddEquivCounitIso AE (AddEquiv1 AE x)) : A2⟦_, _⟧) = # (AddEquiv1 AE) (inv_from_z_iso (AddEquivUnitIso AE x)).
proof: Proof. use (post_comp_with_z_iso_inv_is_inj (AddEquivCounitIso AE (AddEquiv1 AE x))). apply pathsinv0. rewrite (is_inverse_in_precat1 (AddEquivCounitIso AE ((AddEquiv1 AE) x))). rewrite AddEquivCounitUnit. rewrite <- functor_comp. rewrite (is_inverse_in_precat2 (AddEquivUnitIso AE x)). apply functor_id. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.Algebra.Groups UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.BinDirectSums