Definition trivial_adj_equiv_to_disp_adj_equiv {B C : bicat} {b : B} {c₁ c₂ : trivial_displayed_bicat B C b} : adjoint_equivalence c₁ c₂ → disp_adjoint_equivalence (internal_adjoint_equivalence_identity b) c₁ c₂. Proof. intros e. simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))). - exact (pr1 e). - exact (left_adjoint_right_adjoint e). - exact (left_adjoint_unit e). - exact (left_adjoint_counit e). - abstract (unfold transportb ; cbn ; rewrite transportf_const ; cbn ; exact (pr1 (pr122 e))). - abstract (unfold transportb ; cbn ; rewrite transportf_const ; cbn ; exact (pr2 (pr122 e))). - abstract (apply trivial_is_invertible_2cell_to_is_disp_invertible ; apply (pr2 e)). - abstract (apply trivial_is_invertible_2cell_to_is_disp_invertible ; apply (pr2 e)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.Initial. Require Import UniMath.Bicategories.Core.Examples.Final. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.