Lemma presheaf_to_yoneda_ob_pstrans_is_pstrans : is_pstrans presheaf_to_yoneda_ob_pstrans_data. Proof. repeat split. - intros Y₁ Y₂ g₁ g₂ α. apply nat_trans_eq. { apply homset_property. } intro h ; cbn in *. unfold presheaf_to_yoneda_ob_pstrans_functor_ob, presheaf_to_yoneda_ob_pstrans_functor_mor, presheaf_to_yoneda_ob_pstrans_nat_trans_data. pose (psfunctor_lwhisker F h α). pose (nat_trans_eq_pointwise p x) as q. exact (!q). - intros Y. apply nat_trans_eq. { apply homset_property. } intro h ; cbn in *. unfold presheaf_to_yoneda_ob_pstrans_functor_ob, presheaf_to_yoneda_ob_pstrans_functor_mor, presheaf_to_yoneda_ob_pstrans_nat_trans_data. refine (!_). etrans. { etrans. { apply maponpaths_2. apply id_left. } etrans. { apply id_left. } exact (nat_trans_eq_pointwise (psfunctor_rinvunitor F h) x). } cbn -[psfunctor_id psfunctor_comp]. apply maponpaths_2. apply id_left. - intros Y₁ Y₂ Y₃ g₁ g₂. apply nat_trans_eq. { apply homset_property. } intro h ; cbn in *. unfold presheaf_to_yoneda_ob_pstrans_functor_ob, presheaf_to_yoneda_ob_pstrans_functor_mor, presheaf_to_yoneda_ob_pstrans_nat_trans_data. refine (!_). etrans. { etrans. { apply maponpaths_2. etrans. { apply id_right. } apply maponpaths_2. etrans. { apply id_right. } apply id_left. } exact (nat_trans_eq_pointwise (psfunctor_rassociator F h g₁ g₂) x). } simpl. etrans. { apply maponpaths_2. apply id_left. } apply idpath. 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.