Proposition set_rel_double_cat_lunitor_laws : double_lunitor_laws set_rel_double_cat_lunitor_data. Proof. intros X₁ X₂ Y₁ Y₂ R₁ R₂ f g fgR. use funextsec ; intro x. use funextsec ; intro y. use funextsec ; intro p. etrans. { apply transportf_set_rel. } cbn. induction p as [ y' [ p q ]]. induction p. cbn. etrans. { apply maponpaths. use (transportf_set (R₂ (f x))). apply setproperty. } apply maponpaths_2. apply setproperty. Qed.
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.