Definition iso_set_rel_disp_cat {X Y : SET} (R₁ R₂ : set_rel_disp_cat X Y) : (∏ (x : pr1 X) (y : pr1 Y), R₁ x y ≃ R₂ x y) ≃ iso_twosided_disp (identity_z_iso X) (identity_z_iso Y) R₁ R₂. Proof. use weq_iso. - exact (to_iso_set_rel_disp_cat R₁ R₂). - exact (from_iso_set_rel_disp_cat R₁ R₂). - abstract (intros f ; use funextsec ; intro x ; use funextsec ; intro y ; use subtypePath ; [ intro ; apply isapropisweq | ] ; apply idpath). - abstract (intros f ; use subtypePath ; [ intro ; apply isaprop_is_iso_twosided_disp | ] ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Discrete. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedFibration.