Proposition disp_functor_disp_ess_surj_to_split {C : category} {D₁ D₂ : disp_cat C} (HD₁ : is_univalent_disp D₁) (HD₂ : is_univalent_disp D₂) {F : disp_functor (functor_identity C) D₁ D₂} (H₁ : disp_functor_disp_ess_surj F) (H₂ : disp_functor_ff F) : disp_functor_disp_ess_split_surj F. Proof. intros x yy. refine (factor_through_squash _ (λ z, z) (H₁ x yy)). refine (isofhlevelweqf _ _ (isaprop_hfiber_ff_disp HD₁ HD₂ H₂ x yy)). use weqfibtototal. intro xx ; cbn. exact (make_weq _ (HD₂ x x (idpath _) _ _)). Defined.
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.