This might be very basic. Suppose I want to prove that a vector space $V$ with the bilinear map $\kappa : V_1 \times V_2 \rightarrow V$ is a tensor product of $V_1$ and $V_2$. I have to show that the universal property of the tensor product holds, i.e. that for any linear space $W$ and any bilinear map $\phi: V_1 \times V_2 \rightarrow W$ there is a unique linear map $\bar{\phi}: V\rightarrow W$ with $\bar{\phi} \circ \kappa = \phi$. But now, how can I do this in practice?
For example, let $V_1$ and $V_2$ be matrix space of dimensions $m\times n$ and $m'\times n'$ and the tensorproduct given by the kronecker product. Now the universal property seems to be satisfied clearly, but I don't understand the way this can be proven.