Definition disp_left_adjoint_equivalence_subbicat_help {B : bicat} (HB : is_univalent_2 B) (P₀ : B → UU) (P₁ : ∏ (x y : B), x --> y -> UU) (P₁id : ∏ (x : B) (Px : P₀ x), P₁ x x (id₁ x)) (P₁comp : ∏ (x y z : B) (Px : P₀ x) (Py : P₀ y) (Pz : P₀ z) (f : x --> y) (g : y --> z), P₁ _ _ f → P₁ _ _ g → P₁ _ _ (f · g)) {x y : B} (Px : P₀ x) (f : adjoint_equivalence x y) : P₁ x y f. Proof. revert x y f Px. use J_2_0. - exact (pr1 HB). - intro x ; cbn. apply P₁id. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.FullSub. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sigma.