Lemma comp_local_weak_equivalence {B1 B2 B3 : bicat} {F : psfunctor B1 B2} {G : psfunctor B2 B3} (Feso : local_weak_equivalence F) (Geso : local_weak_equivalence G) : local_weak_equivalence (comp_psfunctor G F). Proof. intros x y. split. - use (comp_essentially_surjective (Fmor F x y) _ (Fmor G _ _)). + exact (pr1 (Feso x y)). + apply (pr1 (Geso _ _)). - use (comp_ff_is_ff _ _ _ (Fmor F x y) _ (Fmor G _ _)). + exact (pr2 (Feso x y)). + apply (pr2 (Geso _ _)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.PrecompEquivalence. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.RezkCompletions.RezkCompletions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.YonedaLemma. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Require Import UniMath.Bicategories.PseudoFunctors.Properties. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition.