Definition enrichment_over_smash_struct_weq_struct_enrichment_iso (x y : C) : @z_iso (category_of_hset_struct P) (homset x y ,, make_smash_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) true). + cbn. apply transportf_struct_weq_on_invweq. - simple refine (_ ,, _). + refine (λ f, enriched_to_arr E (pointed_hset_struct_map_from_unit P (pr2 (E ⦃ x, y ⦄)) f ,, _)). apply hset_struct_with_smash_map_from_unit. + 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 b. induction b ; cbn. * apply idpath. * refine (!(pointed_hset_struct_preserve_point P (pr2 (enriched_from_arr E w))) @ _). apply maponpaths. apply hset_struct_with_smash_point_unit. + 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.DisplayedCats.Structures.StructuresSmashProduct. 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.SmashProductMonoidal. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Section FromStructSmashEnrichment. End FromStructSmashEnrichment.