Definition disp_cat_eq_step_5 : (∑ (pob : ∏ (x : C), D₁ x = D₂ x), ∑ (pmor : ∏ (x y : C) (xx : D₁ x) (yy : D₁ y) (f : x --> y), xx -->[ f ] yy = eqweqmap (pob x) xx -->[ f ] eqweqmap (pob y) yy), disp_cat_id_comp_eq pob pmor) ≃ disp_cat_eq_functor. Proof. use weqbandf. - refine (weqonsecfibers _ _ (λ x, _)). exact (univalence _ _). - intro pob. use weqbandf. + cbn. refine (weqonsecfibers _ _ (λ x, _)). refine (weqonsecfibers _ _ (λ y, _)). refine (weqonsecfibers _ _ (λ xx, _)). refine (weqonsecfibers _ _ (λ yy, _)). refine (weqonsecfibers _ _ (λ f, _)). exact (univalence _ _). + cbn. intro pmor. exact (idweq _). 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.