Definition disp_univalent_2_0_enriched_cats_help {C : bicat_of_univ_cats} (E₁ E₂ : disp_bicat_of_enriched_cats C) : enrichment_data_hom_path_help (pr1 E₁) (pr1 E₂) ≃ (∑ (F₁ : functor_enrichment (functor_identity _) E₁ E₂) (F₂ : functor_enrichment (functor_identity _) E₂ E₁), nat_trans_enrichment (nat_trans_id (functor_identity _)) (functor_id_enrichment E₁) (functor_comp_enrichment F₁ F₂) × nat_trans_enrichment (nat_trans_id _) (functor_comp_enrichment F₂ F₁) (functor_id_enrichment E₂) × nat_trans_enrichment (nat_trans_id _) (functor_comp_enrichment F₁ F₂) (functor_id_enrichment E₁) × nat_trans_enrichment (nat_trans_id _) (functor_id_enrichment E₂) (functor_comp_enrichment F₂ F₁)). Proof. use weq_iso. - exact from_enrichment_path. - exact (λ F, to_enrichment_path (pr1 F) (pr12 F) (pr122 F) (pr1 (pr222 F))). - abstract (intros F ; use subtypePath ; [ intro ; repeat (apply isapropdirprod) ; repeat (use impred ; intro) ; apply homset_property | ] ; use funextsec ; intro x ; use funextsec ; intro y ; use z_iso_eq ; apply idpath). - abstract (intros F ; use disp_univalent_2_0_enriched_cats_help_path ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed. Require Import UniMath.CategoryTheory.EnrichedCats.BenabouCosmos. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.QuantaleEnriched. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Projection. Section FromEnrichmentPath. End FromEnrichmentPath.