Definition lax_double_functor_weq_functor_with_lax_extension (V : quantale_cosmos) : lax_double_functor (enriched_rel_univalent_double_cat V) (enriched_rel_univalent_double_cat V) ≃ functor_with_lax_extension V. Proof. use weq_iso. - exact lax_double_functor_to_functor_with_lax_extension. - exact (λ F, lax_extension_to_lax_double_functor (lax_extension_of_functor F)). - exact lax_double_functor_weq_functor_with_lax_extension_left. - exact lax_double_functor_weq_functor_with_lax_extension_right. Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.EnrichedRels. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. 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.DoubleFunctor. Require Import UniMath.Bicategories.DoubleCategories.Examples.EnrichedRelDoubleCat. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichedRelation. Require Import UniMath.CategoryTheory.EnrichedCats.LaxExtension.