Definition disp_presheaves_adjequiv {C : ∁} (FC FC' : disp_presheaf_bicat C) : @invertible_2cell bicat_of_univ_cats _ _ FC FC' -> disp_adjoint_equivalence (internal_adjoint_equivalence_identity C) FC FC'. Proof. intros α. use tpair. - apply α. - use tpair. + use tpair. * apply α. * split ; apply nat_trans_eq ; try (apply homset_property) ; intro x ; cbn. ** rewrite (functor_id FC), id_right. exact (!(nat_trans_eq_pointwise (pr122 α) x)). ** rewrite (functor_id FC'), id_left. exact (nat_trans_eq_pointwise (pr222 α) x). + split ; split. * apply isaset_nat_trans. apply homset_property. * apply isaset_nat_trans. apply homset_property. * apply disp_presheaves_all_invertible. * apply disp_presheaves_all_invertible. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.