Let $X,Y$ be finite sets, $Fun(X),Fun(X),Fun(X\oplus Y)$ be $\Bbb{C}$-modules of functions mapping the corresponding sets to $\Bbb{C}$.
It's obvious that $dim(Fun(X))=|X|$ and ${\{f_{x_0}:X\rightarrow\Bbb{C}, x_0\in\Bbb{C}|f(x_0)=1,f(x)=0, x\neq x_0\}}$ is a basis.
Let consider a homomorphism $\phi:Fun(X)\otimes Fun(Y)\rightarrow Fun(X\oplus Y), f(x)\otimes g(y)\mapsto f(x)\cdot g(y)$. $\phi$ is correctly defined.
To show that $\phi$ is a isomorphism, consider $\phi^{-1}:f(x,y)\mapsto f(x,0)\otimes f(0,y)$.
Is it enough to show that $\phi$ is an isomorphism?