{-# OPTIONS --without-K --safe #-}

module Category.Monoidal.Instance.Nat where

open import Level using (0ℓ)
open import Categories.Category.Monoidal.Bundle using (MonoidalCategory)
open import Categories.Category.Instance.Nat using (Nat; Nat-Cartesian; Nat-Cocartesian; Natop)
open import Categories.Category.Cartesian using (Cartesian)
open import Categories.Category.Cocartesian using (Cocartesian; module CocartesianMonoidal)
open import Categories.Category.Cartesian.Monoidal using (module CartesianMonoidal)
open import Categories.Category.Duality using (coCartesian⇒Cocartesian; Cocartesian⇒coCartesian)

open CartesianMonoidal using () renaming (monoidal to ×-monoidal)
open CocartesianMonoidal using (+-monoidal)
open MonoidalCategory

Natop-Cartesian : Cartesian Natop
Natop-Cartesian = Cocartesian⇒coCartesian Nat Nat-Cocartesian

Natop-Cocartesian : Cocartesian Natop
Natop-Cocartesian = coCartesian⇒Cocartesian Natop Nat-Cartesian

Nat,+,0 : MonoidalCategory 0ℓ 0ℓ 0ℓ
Nat,+,0 .U = Nat
Nat,+,0 .monoidal = +-monoidal Nat Nat-Cocartesian

Nat,×,1 : MonoidalCategory 0ℓ 0ℓ 0ℓ
Nat,×,1 .U = Nat
Nat,×,1 .monoidal = ×-monoidal Nat-Cartesian

Natop,+,0 : MonoidalCategory 0ℓ 0ℓ 0ℓ
Natop,+,0 .U = Natop
Natop,+,0 .monoidal = ×-monoidal Natop-Cartesian

Natop,×,1 : MonoidalCategory 0ℓ 0ℓ 0ℓ
Natop,×,1 .U = Natop
Natop,×,1 .monoidal = +-monoidal Natop Natop-Cocartesian