Definition all_companions_para : all_companions_double_cat para_double_cat. Proof. intros x y f. simple refine (_ ,, _). - exact (para_companion M f). - use make_double_cat_are_companions'. + exact (para_companion_unit M f). + exact (para_companion_counit M f). + abstract (use path_para_sqr ; refine (transportf_disp_mor2_para _ _ _ _ @ _) ; cbn ; rewrite tensor_id_id ; rewrite id_right ; rewrite mon_lunitor_I_mon_runitor_I ; apply mon_rinvunitor_runitor). + abstract (use path_para_sqr ; refine (transportf_disp_mor2_para _ _ _ _ @ _) ; cbn ; apply id_right). Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. 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.ParaTwoSidedDispCat. Require Import UniMath.CategoryTheory.Monoidal.Categories. 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.PseudoDoubleSetCats. Require Import UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints. Require Import UniMath.Bicategories.DoubleCategories.Examples.DoubleCatOnDispCat. Import MonoidalNotations.