Proposition disp_ess_surj_ob_weq {C₁ C₂ : category} {F : C₁ ⟶ C₂} {D₁ : disp_cat C₁} {D₂ : disp_cat C₂} (HD₁ : is_univalent_disp D₁) (HD₂ : is_univalent_disp D₂) {FF : disp_functor F D₁ D₂} (HFF₁ : disp_functor_disp_ess_surj FF) (HFF₂ : disp_functor_ff FF) (x : C₁) : isweq (FF x). Proof. intros yy. use (factor_through_squash _ _ (HFF₁ x yy)). { apply isapropiscontr. } intros xx. induction xx as [ xx i ]. use iscontraprop1. - exact (isaprop_hfiber_ff_disp HD₁ HD₂ HFF₂ x yy). - refine (xx ,, _). exact (isotoid_disp HD₂ (idpath _) i). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Equivalences. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq.