Let's say you have a vector space $V$ and a vector space $W$ over the same field $k$. Then, according to a huge pile of books on representation theory I read, there is an obvious isomorphism $V^\ast\otimes_k W \cong \mathrm{Hom}_k(V,W)$, where $V^\ast=\mathrm{Hom}_k(V,k)$. I concur, it's not hard to write down, but then why don't they just write it down? It is given by \[ \begin{array}{rcl} \phi: V^\ast\otimes_k W &\longrightarrow& \mathrm{Hom}_k(V,W) \\ f\otimes w &\longmapsto& (v\mapsto f(v)w), \end{array}\] when we assume $W$ to be finite-dimensional. Do you want to see the proof?