Definition cod_disp_adj_equiv_to_adj_equiv {c : B} {f₁ f₂ : cod_disp_bicat B c} : disp_adjoint_equivalence (internal_adjoint_equivalence_identity c) f₁ f₂ → ∑ (y : adjoint_equivalence (pr1 f₁) (pr1 f₂)), invertible_2cell (pr2 f₁) (pr1 y · pr2 f₂). Proof. intro e. simple refine (_ ,, _). - simple refine (_ ,, _). + exact (pr11 e). + simpl. simple refine (_ ,, (_ ,, _)). * simple refine (_ ,, (_ ,, _)). ** exact (pr1 (pr112 e)). ** exact (pr11 (pr212 e)). ** exact (pr12 (pr212 e)). * split. ** abstract (refine (maponpaths pr1 (pr1 (pr122 e)) @ _) ; unfold transportb ; cbn ; refine (@pr1_transportf _ (λ _, _) _ _ _ _ _ @ _) ; rewrite transportf_const ; apply idpath). ** abstract (refine (maponpaths pr1 (pr2 (pr122 e)) @ _) ; unfold transportb ; cbn ; refine (@pr1_transportf _ (λ _, _) _ _ _ _ _ @ _) ; rewrite transportf_const ; apply idpath). * split. ** exact (from_is_disp_invertible_2cell_cod _ (pr12 (pr22 e))). ** exact (from_is_disp_invertible_2cell_cod _ (pr22 (pr22 e))). - refine (rinvunitor _ • pr121 e ,, _). is_iso. exact (pr221 e). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. 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.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.Examples.OneTypes. 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. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Trivial. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma.