Linear map between duals induced by linear maps between vector spaces Let $V, W$ be vector spaces over a field $F$ and let $\psi: V \to W$. Show that $\psi$ induces a linear map $\psi^{*}: W^{*} \to V^{*}$ naturally.
Although the question asks for a naturally induced linear map, it does not seem at all that easy to me. Any suggestions will be greatly appreciated!

Hint:
$$\forall\,\,f\in W^*\;\;,\;\;\psi^*(f)(v):=f(\psi(v))$$
