Lemma monoid_z_iso_equiv (X Y : ob monoid_category) : z_iso X Y -> monoidiso X Y. Proof. intro f. use make_monoidiso. - exact (make_weq (pr1 (pr1 f)) (monoid_z_iso_is_equiv X Y f)). - exact (pr2 (pr1 f)). Defined.
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.Monoids. Require Import UniMath.Algebra.Free_Monoids_and_Groups. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Adjunctions.Core.