Lemma monoid_to_category_fully_faithful_functor_iso (f : monoid_to_category_ob M ⟶ monoid_to_category_ob M') : monoid_to_category_mor (functor_to_monoidfun f) = f. Proof. apply (functor_eq _ _ (homset_property _)). use functor_data_eq. - intro t. now induction t, (pr1 f tt). - intros t t' m. do 2 refine (eqtohomot (transportf_const _ _) _ @ _). now induction t, t'. Qed.
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.