Definition adj_equiv_fib_slice_weq (D₁ D₂ : fib_slice_bicat) : (∑ (F : disp_functor (functor_identity C) (pr1 D₁) (pr1 D₂)), is_equiv_over_id F) ≃ adjoint_equivalence D₁ D₂. Proof. use weq_iso. - exact (λ F, adj_equiv_fib_slice (pr1 F) (pr2 F)). - exact (λ F, pr11 F ,, from_left_adjoint_equivalence_fib_slice _ (pr2 F)). - abstract (intros F ; use subtypePath ; [ intro ; apply (isaprop_is_equiv_over_id (pr1 D₁) (pr1 D₂)) | ] ; apply idpath). - abstract (intro F ; use subtypePath ; [ intro ; apply (isaprop_left_adjoint_equivalence _ is_univalent_2_1_fib_slice_bicat) | ] ; use subtypePath ; [ intro ; apply isaprop_is_cartesian_disp_functor | ] ; cbn ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Adjunctions. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence.