Lemma presheaf_to_yoneda_is_functor : is_functor presheaf_to_yoneda_data. Proof. split. - intros z. apply modification_eq. intros Z. apply nat_trans_eq. { apply homset_property. } intros f. cbn in *. unfold presheaf_to_yoneda_mor_modification_nat_trans_data, presheaf_to_yoneda_ob_pstrans_functor_ob. apply functor_id. - intros z₁ z₂ z₃ f₁ f₂. apply modification_eq. intros Z. apply nat_trans_eq. { apply homset_property. } intros f. cbn in *. unfold presheaf_to_yoneda_mor_modification_nat_trans_data, presheaf_to_yoneda_ob_pstrans_functor_ob. apply functor_comp. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Examples.OpMorBicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.Properties. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.UnivalenceOp. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Properties. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.PseudoFunctors.Yoneda. Require Import UniMath.Bicategories.PseudoFunctors.Representable. Require Import UniMath.Bicategories.Core.Examples.Image. Require Import UniMath.Bicategories.PseudoFunctors.Examples.CorestrictImage.