Lemma is_functor_quot_functor_data : is_functor quot_functor_data. Proof. split. - intros a; simpl. apply funextfun. intro c. apply (surjectionisepitosets (setquotpr _)); [now apply issurjsetquotpr | apply isasetsetquot|]. intro x; cbn. now rewrite (functor_id R). - intros a b c f g; simpl. apply funextfun; intro x. apply (surjectionisepitosets (setquotpr _)); [now apply issurjsetquotpr | apply isasetsetquot|]. intro y; cbn. now rewrite (functor_comp R). Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Epis. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.EpiFacts. Require Import UniMath.CategoryTheory.Limits.Coequalizers.