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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?

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up vote 7 down vote accepted

The functor $F$ is a contravariant functor, that is why it reverses composition. Check out the definitions at

http://en.wikipedia.org/wiki/Functor

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Thanks for pointing this out. –  Jonah777 Sep 14 '12 at 22:41
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