Proposition cartesian_semi_cart_tensor_pr1 (VC : category) (TV : Terminal VC) (VP : BinProducts VC) (expV : Exponentials VP) (V : sym_mon_closed_cat := sym_mon_closed_cartesian_cat VC VP TV expV) (x y : V) : semi_cart_tensor_pr1 (is_semicartesian_cartesian_monoidalcat VC VP TV) x y = BinProductPr1 _ _. Proof. unfold semi_cart_tensor_pr1 ; cbn. unfold monoidal_cat_tensor_mor, functoronmorphisms1 ; cbn. rewrite !assoc'. rewrite BinProductOfArrowsPr1. rewrite id_right. rewrite BinProductOfArrowsPr1. apply id_right. 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.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.