Proposition profunctor_runitor_laws : double_runitor_laws profunctor_runitor_data. Proof. intros C₁ C₂ D₁ D₂ P Q F₁ F₂ τ. use eq_profunctor_square. intros y x h. etrans. { apply transportf_disp_mor2_profunctors. } cbn in *. use (mor_from_comp_profunctor_ob_eq P (id_profunctor _) y x (λ h, τ y x (runitor_profunctor_nat_trans_mor P y x h)) (λ h, runitor_profunctor_nat_trans_mor Q (F₂ y) (F₁ x) (comp_v_profunctor_square_mor τ (id_v_profunctor_square _) y x h))). clear h. intros z h h' ; cbn. refine (!_). etrans. { apply maponpaths. apply (comp_v_profunctor_square_mor_comm τ (id_v_profunctor_square _)). } rewrite !runitor_profunctor_nat_trans_mor_comm ; cbn. rewrite (profunctor_square_lmap τ). 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.Categories.HSET.All. Require Import UniMath.CategoryTheory.Categories.HSET.SetCoends. Require Import UniMath.CategoryTheory.Profunctors.Core. Require Import UniMath.CategoryTheory.Profunctors.Examples. Require Import UniMath.CategoryTheory.Profunctors.Squares. Require Import UniMath.CategoryTheory.Profunctors.Transformation. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.ProfunctorTwosidedDispCat. 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.CompanionsAndConjoints.