proposition: Lemma fully_faithful_inv_identity (C D : precategory_data) (F : functor C D) (FF : fully_faithful F) (a : ob C) : FF^-1 (identity (F a)) = identity _.
proof: Proof. apply (invmaponpathsweq (weq_from_fully_faithful FF a a)). unfold fully_faithful_inv_hom. set (HFaa:=homotweqinvweq (weq_from_fully_faithful FF a a)(identity _ )). simpl in *. rewrite HFaa. rewrite functor_id; apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.PartA UniMath.MoreFoundations.Univalence UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.TransportMorphisms UniMath.CategoryTheory.Core.Univalence