Proposition disp_locally_groupoid_disp_bicat_comp_cat : disp_locally_groupoid disp_bicat_comp_cat. Proof. use make_disp_locally_groupoid_univalent_2_1. - refine (λ (C₁ C₂ : cat_with_terminal_cleaving) (F : functor_with_terminal_cleaving C₁ C₂) (χ₁ : comprehension_functor C₁) (χ₂ : comprehension_functor C₂) (Fχ₁ Fχ₂ : comprehension_nat_trans χ₁ χ₂ F) (p : comprehension_nat_trans_eq (id₂ _) Fχ₁ Fχ₂), _). simple refine (_ ,, _ ,, _) ; try (apply disp_2cells_isaprop_disp_bicat_comp_cat). intros x xx ; cbn. rewrite id_left. etrans. { apply maponpaths. apply comprehension_functor_mor_id. } rewrite id_right. pose (q := p x xx). cbn in q. rewrite id_left in q. refine (!q @ _). etrans. { apply maponpaths. apply comprehension_functor_mor_id. } apply id_right. - exact is_univalent_2_1_bicat_cat_with_terminal_cleaving. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. 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.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.ComprehensionC. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal. 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.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.