Describe the square matrix that represents $K^{mn} \cong K^{m}\otimes_{K}K^{n}\xrightarrow[]{{f\otimes g}}K^{m}\otimes_{K}K^{n}\cong K^{mn}$

Question

Let $K$ be field and let $m$ and $n$ be natural numbers. We have have seen, that the isomorphism
$K^{m}\otimes_{K}K^{n} \cong K^{mn}$, is given by $e_i\otimes e'_j :=\widetilde{e}_{(i-1)n+j}$.

The natural basis of $K^{m}$ is here given by $e_1, \ldots,e_m$ and the natural basis of $K^n$ by $e'_1,\ldots,e'_n$.

The natural basis of $K^{mn}$ is given by $\widetilde{e}_1,\ldots,\widetilde{e}_{mn}$.

Now let $A:= (a_{ij}) \in \ M_m(K) $ and $B:= (b_{ij}) \in \ M_n(K)$ be square matrices.

Let $f:K^{m} \Rightarrow K^{m}$ ; $f(x) := Ax $ and $g:K^{n} \Rightarrow K^{n}$; $g(y) := By$ be the corresponding endomorphisms.

Describe the square matrix that represents in regards to the natural basis of $K^{mn}$ the endomorphism given by:

$K^{mn} \cong K^{m}\otimes_{K}K^{n}\xrightarrow[]{{f\otimes g}}K^{m}\otimes_{K}K^{n}\cong K^{mn}$.

I'm really having trouble figuring this one out. Any help is greatly appreciated :)