TITLE: Sending vector space to dual is a functor
QUESTION [6 upvotes]: In the category of finite dimensional vector spaces over a field and linear maps between them, the map that sends each space to its dual and linear map to its transpose is a functor, right?
But this doesn't make sense to me.  Call the map $F$, and let $f: V \to W$ and $g: W \to Z$ be linear maps.  So,
$$ F(g \circ f) = (g \circ f)^{*} :Z^* \to V^*$$
needs to be  $F(g) \circ F(f)$ but this is not defined, $F(f) \circ F(g)$ is defined however.  Can someone tell me what I'm missing here?

REPLY [10 votes]: The functor $F$ is a contravariant functor, that is why it reverses composition. Check out the definitions at
http://en.wikipedia.org/wiki/Functor