Definition monoid_iso_weq_monoid_category_equiv : ∏ (M M' : monoid), z_iso (C := monoid_category) M M' ≃ z_iso (C := cat_of_setcategory) (monoid_to_category_ob M) (monoid_to_category_ob M') := weq_ff_functor_on_z_iso monoid_to_cat_fully_faithful.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Monoids. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.Categories.Monoid. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.AlgebraicTheories.FundamentalTheorem.CommonUtilities.MonoidActions.