Lemma weak_equivalence_to_is_weak_biequivalence {B₁ B₂ : bicat} {HB₁ : is_univalent_2_1 B₁} {HB₂ : is_univalent_2_1 B₂} (F : psfunctor B₁ B₂) (HF : weak_equivalence HB₁ HB₂ F) : weak_biequivalence F. Proof. exists (pr2 HF). intros x y. split. - apply functor_from_equivalence_is_essentially_surjective. apply (adj_equiv_to_equiv_cat (Fmor_univ F x y HB₁ HB₂)). exact (pr1 HF x y). - apply fully_faithful_from_equivalence. apply (adj_equiv_to_equiv_cat (Fmor_univ F x y HB₁ HB₂)). exact (pr1 HF x y). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations.