proposition: Lemma is_functor_quot_functor_data : is_functor quot_functor_data.
proof: 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.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.MonoEpiIso UniMath.CategoryTheory.Categories.HSET.Colimits UniMath.CategoryTheory.Categories.HSET.Structures UniMath.CategoryTheory.Epis UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.EpiFacts UniMath.CategoryTheory.Limits.Coequalizers