Definition cat_with_equalizers_has_RezkCompletion : disp_left_universal_arrow LUR (disp_psfunctor_on_cat_to_univ_cat disp_bicat_have_equalizers (disp_2cells_isaprop_from_disp_2cells_iscontr _ disp_2cells_is_contr_have_equalizers)). Proof. use make_disp_left_universal_arrow_if_contr_CAT_from_weak_equiv. - exact η_weak_equiv. - intros C1 C2 C2_univ F Fw [E₁ _]. exact (weak_equiv_into_univ_creates_hasequalizers C2_univ Fw E₁ ,, tt). - intros C ?. refine (tt ,, weak_equiv_preserves_equalizers (η_weak_equiv C)). - intros C1 C2 C3 F G H α E₁ E₂ E₃ Gw [t Feq]. exact (tt ,, weak_equiv_lifts_preserves_equalizers C2 C3 α Gw Feq). 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.Equalizers. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.WeakEquivalences.Preservation.Equalizers. Require Import UniMath.CategoryTheory.WeakEquivalences.Reflection.Equalizers. Require Import UniMath.CategoryTheory.WeakEquivalences.Creation.Equalizers. Require Import UniMath.CategoryTheory.WeakEquivalences.LiftPreservation.Equalizers. 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.DisplayedBicats.DispPseudofunctor. 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.