Proposition is_univalent_disp_disp_cat_of_strict_twosided_disp_cat : is_univalent_disp disp_cat_of_strict_twosided_disp_cat. Proof. use is_univalent_disp_from_fibers. intros C D₁ D₂. cbn in C, D₁, D₂. use weqhomot. - refine (_ ∘ path_sigma_hprop _ _ _ (isaprop_is_strict_twosided_disp_cat _))%weq. refine (_ ∘ make_weq _ (isweqmaponpaths (two_sided_disp_cat_weq_disp_cat C C) D₁ D₂))%weq. refine (_ ∘ disp_cat_eq _ _)%weq. use weqtotal2. + exact (invweq (two_sided_disp_functor_weq_disp_functor (functor_identity C) (functor_identity C) D₁ D₂) ∘ disp_functor_over_pair_weq _ _)%weq. + intro F. apply is_z_iso_disp_strict_twosided_disp_cat_weq. - intro p. induction p. use subtypePath. { intro. apply isaprop_is_z_iso_disp. } use subtypePath. { intro. apply isaprop_twosided_disp_functor_laws. } apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. 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.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.DispFunctorPair. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws. Section FromIso. End FromIso.