Lemma locally_univalent_lemma (x y : B) : is_univalent (R (hom (C := B) x y)) -> is_univalent (hom (C := LRB) x y). Proof. intro u. intros f g. assert (p : (λ p : f = g, @idtoiso (R (@hom B x y)) f g p) = (λ p : f = g, @idtoiso (@hom LRB x y) f g p)). { apply funextsec ; intro p. induction p. use total2_paths_f. 2: apply isaprop_is_z_isomorphism. apply idpath. } induction p. apply u. Qed.
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.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.PrecompEquivalence. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.RezkCompletions.RezkCompletions. 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.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.YonedaLemma. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Properties. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.