Definition disp_cat_eq_step_7 : (∑ (F : disp_functor (functor_identity C) D₁ D₂), (∏ (x : C), isweq (F x)) × disp_functor_ff F) ≃ ∑ (F : disp_functor (functor_identity C) D₁ D₂), disp_functor_disp_ess_surj F × disp_functor_ff F. Proof. use weqfibtototal. intro FF. use weqimplimpl. - intros HFF. split. + intros x yy. apply hinhpr. refine (invmap (make_weq _ (pr1 HFF x)) yy ,, _). apply (idtoiso_disp (idpath _)) ; cbn. apply (homotweqinvweq (make_weq (FF x) (pr1 HFF x))). + exact (pr2 HFF). - intros HFF. split. + intro x. exact (disp_ess_surj_ob_weq HD₁ HD₂ (pr1 HFF) (pr2 HFF) x). + exact (pr2 HFF). - use isapropdirprod. + use impred ; intro. apply isapropisweq. + apply isaprop_disp_functor_ff. - apply isapropdirprod. + apply propproperty. + apply isaprop_disp_functor_ff. 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.