Definition bicategorical_yoneda_lemma : left_adjoint_equivalence yoneda_to_presheaf. Proof. apply equiv_to_isadjequiv. use tpair. - use tpair. + exact presheaf_to_yoneda. + split. * exact yoneda_unit. * exact yoneda_counit. - split. + cbn. apply is_nat_z_iso_to_is_invertible_2cell. intro g. apply is_inv2cell_to_is_z_iso. apply make_is_invertible_modification. intro Z. apply is_nat_z_iso_to_is_invertible_2cell. intros Y. exact (yoneda_unit_z_iso g Z Y). + apply is_nat_z_iso_to_is_invertible_2cell. intros Z ; cbn. unfold yoneda_counit_component. exists (pr1 (pr1 (psfunctor_id F X)) Z). split. * abstract (exact (nat_trans_eq_pointwise (vcomp_linv (psfunctor_id F X)) Z)). * abstract (exact (nat_trans_eq_pointwise (vcomp_rinv (psfunctor_id F X)) Z)). Defined.
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.