Definition set_rel_double_cat_runitor_data : double_runitor_data set_rel_double_cat_hor_id set_rel_double_cat_hor_comp. Proof. refine (λ (X Y : hSet) (R : X → Y → hSet), _). use to_iso_set_rel_disp_cat. intros x y ; cbn. use weq_iso. - exact (λ p, transportf (λ z, R x z) (pr22 p) (pr12 p)). - exact (λ p, y ,, p ,, idpath _). - abstract (intros p ; induction p as [ x' [ p q ]] ; induction q ; apply idpath). - abstract (intro ; apply idpath). Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Relations. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Categories.Rel. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.SymmetricUnivalent. Require Import UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints.