proposition: 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: 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.
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