If $\Phi: \mathbf{Vec} \rightarrow \mathbf{Vec}$ with $\Phi(V) = V^{\ast\ast}$ and $f: V \rightarrow W$, what is $\Phi(f)$?

Question

Let $\Phi$ be an endofunctor of the category of vector spaces over a field which sends a vector space to its double dual. Let $V$ and $W$ be 2 vector spaces and let $f: V \rightarrow W$ be a morphism from $V$ to $W$. Then $\Phi(f): V^{\ast\ast} \rightarrow W^{\ast\ast}$, but do I know how $\Phi(f)$ acts on elements of $V^{\ast\ast}$?

Answer 1

By definition, $V^*=\hom(V,K)$ where $K$ is the base field.

Well, $\Phi(f)$ assigns for an element $\vartheta\in V^{**}=\hom(V^*,K)$ the element of $W^{**}=\hom(W^*,K)$ which sends $$\underset{W\to K}g\mapsto \vartheta\big(\underset{V\to K}{g\circ f}\big)\,.$$