Proposition pointed_struct_pointed_hset_with_smash_adj_laws : hset_struct_with_smash_closed_adj_laws pointed_struct_pointed_hset_with_smash_closed_pointed. Proof. split. - intros X Y Z f. induction X as [ X x ]. induction Y as [ Y y ]. induction Z as [ Z z ]. induction f as [ f p ]. cbn in *. exact (eqtohomot (maponpaths pr1 p) y). - intros X Y Z f. induction X as [ X x ]. induction Y as [ Y y ]. induction Z as [ Z z ]. induction f as [ f p ]. use subtypePath. { intro. apply setproperty. } use funextsec. intro a. refine (_ @ p). refine (maponpaths f _). use iscompsetquotpr. apply hinhpr ; cbn. use inr. unfold product_point_coordinate ; cbn. unfold pointed_struct_pointed_hset_data. split ; use inl ; apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits. Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructuresSmashProduct.