Let $G$ be a finite group. For any finite $G$-set $X$, let $$\mathbb{C}\left<X\right> := \left\{\sum_{x \in X} c_x x \mid c_x \in \mathbb{C} \right\}. $$ Now for finite $G$-sets $Y, Z$, consider the $G$-set $Y \times Z$ by $g(y,z) = (gy, gz)$.
I want to show that there exists a canonical isomorphism
$$\mathbb{C}\left<Y\right> \otimes_{\mathbb{C}} \mathbb{C}\left<Z\right> \tilde{\longrightarrow} \mathbb{C}\left<Y \times Z\right> $$ of representations of $G$.
Normally we do this by exploiting the universal property of the tensor product: $\mathbb{C}\left<Y\right> \otimes_{\mathbb{C}} \mathbb{C}\left<Z\right>$ together with a $\mathbb{C}$-bilinear map $t : \mathbb{C}\left<Y\right> \times \mathbb{C}\left<Z\right> \longrightarrow \mathbb{C}\left<Y\right> \otimes_{\mathbb{C}} \mathbb{C}\left<Z\right> $ forms the tensor product of $\mathbb{C}\left<Y\right>$ and $\mathbb{C}\left<Z\right>$. For every Abelian group $A$ and $\mathbb{C}$-bilinear map $b : \mathbb{C}\left<Y\right> \times \mathbb{C}\left<Z\right> \longrightarrow A$ there is a unique group homomorphism $\tilde{b} : \mathbb{C}\left<Y\right> \otimes_{\mathbb{C}} \mathbb{C}\left<Z\right> \longrightarrow A$ such that $\tilde{b} \circ t = b$.
Now if we could show that in fact $\mathbb{C}\left< Y\times Z\right>$ is itself a tensor product in this way of $\mathbb{C}\left<Y\right>$ and $\mathbb{C}\left<Z\right>$, we could obtain an inverse group homomorphism of $\tilde{b}$, proving the desired isomorphism. I'm new to this whole kind of reasoning, so it would help me a lot to see how one goes about this.