Library UniMath.CategoryTheory.Monoidal.Examples.StructuresMonoidal

Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors.
Require Import UniMath.CategoryTheory.Monoidal.Categories.
Require Import UniMath.CategoryTheory.Monoidal.Structure.Cartesian.
Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric.
Require Import UniMath.CategoryTheory.Monoidal.Structure.Closed.
Require Import UniMath.CategoryTheory.Monoidal.Examples.CartesianMonoidal.
Require Import UniMath.CategoryTheory.DisplayedCats.Structures.CartesianStructure.
Require Import UniMath.CategoryTheory.DisplayedCats.Structures.StructureLimitsAndColimits.
Require Import UniMath.CategoryTheory.limits.binproducts.
Require Import UniMath.CategoryTheory.limits.terminal.

Definition monoidal_cat_of_hset_struct
(P : hset_cartesian_struct)
: monoidal_cat.
Show proof.

  refine (category_of_hset_struct P ,, _).
  use cartesian_monoidal.
  - exact (BinProducts_category_of_hset_struct P).
  - exact (Terminal_category_of_hset_struct P).

Proposition is_cartesian_cat_of_hset_struct
(P : hset_cartesian_struct)
: is_cartesian (monoidal_cat_of_hset_struct P).
Show proof.

apply is_cartesian_cartesian_monoidalcat.

Definition sym_monoidal_cat_of_hset_struct
           (P : hset_cartesian_struct)
  : sym_monoidal_cat
  := monoidal_cat_of_hset_struct P ,, symmetric_cartesian_monoidalcat _ _ _.

Definition sym_mon_closed_cat_of_hset_struct
           (P : hset_cartesian_closed_struct)
  : sym_mon_closed_cat.
Show proof.

  use sym_mon_closed_cartesian_cat.
  - exact (category_of_hset_struct P).
  - exact (BinProducts_category_of_hset_struct P).
  - exact (Terminal_category_of_hset_struct P).
  - exact (Exponentials_struct P).