Definition monoid_action_to_presheaf_ob (X : monoid_action M) : PreShv (monoid_to_category_ob M). Proof. use make_functor. - use make_functor_data. + intro. exact (X : hSet). + intros a b m x. exact (action x m). - abstract ( split; [ intro t; apply funextfun; intro x; apply action_unel | intros t t' t'' m m'; apply funextfun; intro x; apply action_op ] ). Defined.
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.