Lemma yoneda_fully_faithful (C : category) : fully_faithful (yoneda C). Proof. intros a b; simpl. apply (isweq_iso _ (yoneda_map_1 C a (pr1 (yoneda C) b))). - intro; simpl in *. apply id_left. - intro gamma. simpl in *. apply nat_trans_eq. apply (has_homsets_HSET). intro x. simpl in *. apply funextsec; intro f. unfold yoneda_map_1. unfold yoneda_morphisms_data. assert (T:= toforallpaths _ _ _ (nat_trans_ax gamma a x f) (identity _ )). cbn in T. eapply pathscomp0; [apply (!T) |]. apply maponpaths. apply id_right. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.whiskering.