Suppose $F$ is a field, $A$,$B$ are finite dimensional algebras, $V$ and $W$ finite dimensional representations of $A$ and $B$ respectively, we have the following map
$$ End_A V \otimes End_B W \to End_{A \otimes B} {V \otimes W} $$
It is injective since it is a restriction of the case where the algebras are both $F$, but when is it surjective? It is probably true in algebraically closed fields with semisimple algebras, but are these conditions necessary?