Definition make_disp_univ_arrow_if_prop_equiv {x : B2} (xx : D2 x) {y : B1} (yy : D1 y) : is_equiv_over (left_universal_arrow_functor R x y (pr12 LUR),, (pr22 LUR) x y) (disp_left_universal_arrow_functor LUR RR (LL,, ηη) xx yy). Proof. simple refine (((_ ,, (_,,_)) ,, _) ,, (_ ,, _)). - exact (make_disp_functor_if_prop xx yy). - simple refine (_ ,, _). + exact (LL_unit x xx y yy). + abstract (intro ; intros ; apply D1_2cells_prop). - simple refine (_ ,, _). + exact (LL_counit x xx y yy). + abstract ( intro ; intros ; apply D2_2cells_prop). - abstract (split ; intro ; intros; [ apply D2_2cells_prop | apply D1_2cells_prop ]). - intros f ff. simple refine (_ ,, (_ ,, _)). + exact (LL_unit_inv x xx y yy f ff). + abstract (apply D1_2cells_prop). + abstract (apply D1_2cells_prop). - intros f ff. simple refine (_ ,, (_ ,, _)). + exact (LL_counit_inv x xx y yy f ff). + abstract (apply D2_2cells_prop). + abstract (apply D2_2cells_prop). 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.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Adjunctions. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.TotalAdjunction. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispPseudofunctor. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.PseudoFunctors.Biadjunction. Require Import UniMath.Bicategories.Modifications.Modification. Require Import UniMath.Bicategories.PseudoFunctors.UniversalArrow. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispBiadjunction. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Import DispBicat.Notations.