Definition to_iso_rel_disp_cat {X Y : SET} (R₁ R₂ : rel_disp_cat X Y) (f : ∏ (x : pr1 X) (y : pr1 Y), (R₁ x y → R₂ x y) × (R₂ x y → R₁ x y)) : iso_twosided_disp (identity_z_iso X) (identity_z_iso Y) R₁ R₂. Proof. simple refine ((λ x y, pr1 (f x y)) ,, (λ x y, pr2 (f x y)) ,, _ ,, _). - abstract (cbn ; use funextsec ; intro x ; use funextsec ; intro y ; use funextsec ; intro r ; apply R₁). - abstract (cbn ; use funextsec ; intro x ; use funextsec ; intro y ; use funextsec ; intro r ; apply R₂). 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.