Definition sym_mon_closed_cartesian_cat (expC : Exponentials CP) : sym_mon_closed_cat. Proof. use make_sym_mon_closed_cat. - exact (cartesian_monoidalcat ,, symmetric_cartesian_monoidalcat). - exact (λ x, exp (expC x)). - exact (λ x, exp_eval_alt (expC x)). - exact (λ _ _ _ f, exp_lam_alt (expC _) f). - abstract (cbn ; unfold monoidal_cat_tensor_mor ; unfold functoronmorphisms1 ; cbn ; intros x y z f ; refine (_ @ exp_beta_alt (expC _) f) ; apply maponpaths_2 ; apply prod_lwhisker_rwhisker). - abstract (intros x y z f ; cbn in * ; unfold monoidal_cat_tensor_mor ; unfold functoronmorphisms1 ; cbn ; refine (exp_eta_alt (expC _) f @ _) ; apply maponpaths ; apply maponpaths_2 ; refine (!_) ; apply prod_lwhisker_rwhisker). 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.exponentials. 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.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Import BifunctorNotations. Import MonoidalNotations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core.