In my understanding, the definition of tensor product of linear maps cannot be directly derived from the definition of tensor product of vector spaces (or modules), since it's not clear what is the domain, range, and the map of the result product. Originally I thought this is just a definition. But then I learned that if we represent $S$ and $T$ by matrices, then the matrix describing the tensor product $S \otimes T$ is the Kronecker product of the two matrices.
However, Kronecker product of matrices can be developed totally without the concept of tensor product of linear maps. For example, we can find basis of the matrix spaces (say $V$ and $W$). Then we can get natural basis for $V \otimes W$. And we can easily get the definition of Kronecker product from these basis.
So my question is: is this coincidence just a coincidence or there is some deep reason behind the definition so that it has to be defined like this. I can't even see why the domain of $S \otimes T$ should be $V\otimes W$, given $S$ and $T$ are linear maps over $V$ and $W$.
(I know that the definition of tensor product of linear maps is very natural, and I can't imagine other definitions of it. I just thought there must be a formal reason that the definition has to be that. )