Definition enrichment_over_struct_weq_struct_enrichment_iso (x y : C) : @z_iso (category_of_hset_struct P) (homset x y ,, make_struct_enrichment_data C E x y) (E ⦃ x , y ⦄). Proof. use make_z_iso. - simple refine (_ ,, _). + exact (λ f, pr1 (enriched_from_arr E f) tt). + cbn. apply transportf_struct_weq_on_invweq. - simple refine (_ ,, _). + refine (λ f, enriched_to_arr E ((λ _, f) ,, _)). apply hset_struct_const. + cbn. apply transportf_struct_weq_on_weq. - split. + use eq_mor_hset_struct. intro w ; cbn. refine (_ @ enriched_to_from_arr E _). apply maponpaths. use eq_mor_hset_struct. intro t. cbn. apply maponpaths. apply isapropunit. + use eq_mor_hset_struct. intro f. cbn. rewrite enriched_from_to_arr. 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.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian. Require Import UniMath.CategoryTheory.Monoidal.Examples.StructuresMonoidal. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Section FromStructEnrichment. End FromStructEnrichment.