Definition cat_with_chosen_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). set (t := weak_equiv_preserves_chosen_terminal _ Fw (pr1 C1_term)). exact (F (pr11 C1_term) ,, t). - intros C C1_term. refine (tt ,, _). apply hinhpr. apply idpath. - intros C1 C2 C3 F G H α C1_term C2_term C3_term Gw. intros [t Fpterm]. exists tt. use (factor_through_squash _ _ Fpterm). { apply isapropishinh. } intro p. set (Gpterm := weak_equiv_preserves_chosen_terminal_eq _ Gw (pr2 C2) (pr1 C1_term) (pr1 C2_term)). use (factor_through_squash _ _ Gpterm). { apply isapropishinh. } intro q. apply hinhpr. refine (_ @ p). etrans. { apply maponpaths, (! q). } apply C3. simpl in C1_term ; exact (_ ,, pr2 α (pr1 C1_term)). 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.