Definition cat_with_terminal_obj_has_RezkCompletion : disp_left_universal_arrow LUR RR. Proof. use make_disp_left_universal_arrow_if_contr_CAT_from_weak_equiv. - exact η_weak_equiv. - intros C1 C2 C2_univ F Fw C1_term. refine (_ ,, tt). use (factor_through_squash _ _ (pr1 C1_term)). { apply isapropishinh. } clear C1_term. intro C1_term. apply hinhpr. set (t := weak_equiv_preserves_chosen_terminal _ Fw C1_term). exact (F C1_term ,, t). - intros C C1_term. refine (tt ,, _). use (factor_through_squash _ _ (pr1 C1_term)). { apply isaprop_preserves_terminal. } intro C1_term'. use weak_equiv_preserves_terminal. apply η_weak_equiv. - simpl ; intros C1 C2 C3 F G H α [T1 _] [T2 _] [T3 _] Gw. intros [t Fpterm]. exists tt. exact (weak_equiv_lifts_preserves_terminal F G H α Gw Fpterm). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.WeakEquivalences.Core. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.WeakEquivalences.Terminal. 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.Examples.BicatOfCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.PseudoFunctors.UniversalArrow. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.BicatOfCatToUnivCat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DispBicatOnCatToUniv. Require Import UniMath.Bicategories.DisplayedBicats.Examples.CategoriesWithStructure. Require Import UniMath.Bicategories.DisplayedBicats.DisplayedUniversalArrow. Require Import UniMath.Bicategories.DisplayedBicats.DisplayedUniversalArrowOnCat.