proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Core.EquivToAdjequiv UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor