Definition disp_presheaves_adjequiv_inv {C : ∁} (FC FC' : disp_presheaf_bicat C) : disp_adjoint_equivalence (internal_adjoint_equivalence_identity C) FC FC' → @invertible_2cell bicat_of_univ_cats _ _ FC FC'. Proof. intros α. use tpair. - apply α. - use tpair. + apply α. + split. * apply nat_trans_eq. { apply homset_property. } intro x ; cbn. pose (nat_trans_eq_pointwise (pr1(pr212 α)) x) as p. cbn in p. rewrite (functor_id FC), id_right in p. exact (!p). * apply nat_trans_eq. { apply homset_property. } intro x ; cbn. pose (nat_trans_eq_pointwise (pr2(pr212 α)) x) as p. cbn in p. rewrite (functor_id FC'), id_right in p. exact p. 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.