Definition disp_cat_eq_step_8 : (∑ (F : disp_functor (functor_identity C) D₁ D₂), disp_functor_disp_ess_surj F × disp_functor_ff F) ≃ ∑ (F : disp_functor (functor_identity C) D₁ D₂), is_equiv_over_id F. Proof. use weqfibtototal. intro F. use weqimplimpl. - intros HF. use is_equiv_from_ff_ess_over_id. + exact (disp_functor_disp_ess_surj_to_split HD₁ HD₂ (pr1 HF) (pr2 HF)). + exact (pr2 HF). - intros HF. split. + exact (is_equiv_over_id_to_ess_surj F HF). + exact (is_equiv_over_id_to_ff F HF). - apply isapropdirprod. + apply propproperty. + apply isaprop_disp_functor_ff. - exact (isaprop_is_equiv_over_id HD₁ HD₂ F). 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.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.EquivalenceOverId.