Proposition make_enrichment_over_smash_struct_laws : enrichment_laws make_enrichment_over_smash_struct_data. Proof. repeat split. - intros x y. use eq_mor_hset_struct. use setquotunivprop' ; [ intro ; apply homset_property | ]. intro a. induction a as [ b f ]. induction b ; cbn in *. + rewrite id_right. apply idpath. + rewrite (smash_struct_enrichment_comp_point_l SC). apply idpath. - intros x y. use eq_mor_hset_struct. use setquotunivprop' ; [ intro ; apply homset_property | ]. intro a. induction a as [ f b ]. induction b ; cbn in *. + rewrite id_left. apply idpath. + rewrite (smash_struct_enrichment_comp_point_r SC). apply idpath. - intros w x y z. use eq_mor_hset_struct. use setquotunivprop' ; [ intro ; apply homset_property | ]. intro a. induction a as [ a h ]. revert a. use setquotunivprop' ; [ intro ; apply homset_property | ]. intro a. induction a as [ f g ] ; cbn in *. rewrite assoc. apply idpath. - intros x y f. use eq_mor_hset_struct. intro a. induction a ; cbn. + apply idpath. + rewrite <- (pointed_hset_struct_preserve_point _ (pr2 f)). apply maponpaths. apply hset_struct_with_smash_point_unit. Qed.
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.