Proposition cartesian_tensor_pr1 {x₁ x₂ y₁ y₂ : M} (f : x₁ --> x₂) (g : y₁ --> y₂) : f #⊗ g · semi_cart_tensor_pr1 M_cart x₂ y₂ = semi_cart_tensor_pr1 M_cart x₁ y₁ · f. Proof. unfold semi_cart_tensor_pr1. rewrite !assoc'. rewrite <- tensor_runitor. rewrite !assoc. apply maponpaths_2. rewrite <- !tensor_comp_mor. rewrite !id_left, !id_right. apply maponpaths. apply (semi_cart_to_unit_eq M_cart). 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.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Structure.Symmetric. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Import BifunctorNotations. Import MonoidalNotations.