Proposition lax_double_functor_weq_functor_with_lax_extension_left {V : quantale_cosmos} (F : lax_double_functor (enriched_rel_univalent_double_cat V) (enriched_rel_univalent_double_cat V)) : lax_extension_to_lax_double_functor (lax_double_functor_to_lax_extension F) = F. Proof. use subtypePath. { intro. apply isapropunit. } use subtypePath. { intro. repeat (use isapropdirprod) ; repeat (use impred ; intro) ; apply isasetaprop ; apply isaprop_enriched_relation_square. } use subtypePath. { intro. use isapropdirprod. - use isaproptotal2. { intro. apply isaprop_double_functor_hor_id_laws. } intros. repeat (use funextsec ; intro). apply (is_poset_category_quantale_cosmos V). - use isaproptotal2. { intro. apply isaprop_double_functor_hor_comp_laws. } intros. repeat (use funextsec ; intro). apply (is_poset_category_quantale_cosmos V). } use total2_paths_f. - apply idpath. - use subtypePath. { intro. apply isaprop_twosided_disp_functor_laws. } cbn. use subtypePath. { intro. repeat (use impred ; intro). apply V. } apply idpath. Qed.
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.