# Examples of natural transformations (part 2)

Here’s the most common example of a natural transformation that I know of, and probably the most enlightening. Take a vector space $V$ and let $V^*$ denote it’s dual. If $V$ is finite dimensional, then $V\cong V^*$, but not in any “natural” way. Moreover, even if $V$ is infinite dimensional, $V$ embeds in $V^*$, but again, not in a “natural” way. That is, we have to pick a basis to show how $V$ embeds in $V^*$. If we pick a different basis, we get a different embedding. We don’t have any method that is inherent to the structure. And it’s not just that we haven’t found one. They don’t exist.

However, $V$ does embed inside $V^{**}$ naturally (i.e., in a way independent of the structure of $V$. How are we to describe this? We say that there is a natural transformation between the functors $\mbox{id}_{\textsc{Vec}}:\textsc{Vec}\to\textsc{Vec}$ and $-^{**}:\textsc{Vec}\to\textsc{Vec}$. I’m suppressing the field over which we’re working in the notation. Oh well. It’s not too important.

So here goes. Let $\eta_V:V\to V^{**}$ by $\eta_V(v)=\mbox{ev}_v$. By $\mbox{ev}_v$ what I mean is the map that takes in a functional $\phi$ from $V^*$ and evaluates it at $v$. That is $\mbox{ev}_v:\phi\mapsto \phi(v)$. Since $\phi\in V^*$, and $\mbox{ev}_v$ is a map from $V^*$ to the underlying field, $\mbox{ev}_v$ is an element of $V^{**}$. Does this make

commute?

If we go across the top and then down, we take $v\in V$ and send it to

$T^{**}\circ\eta_V(v)=T^{**}\circ \mbox{ev}_v=T**(\phi\mapsto \phi(v))$

$=(\phi\mapsto (T^*\circ\phi)(v))=(\phi\mapsto\phi(Tv))=\mbox{ev}_{Tv}$.

If we go down and then across, we take $v\in V$ and send it to

$(\eta_W\circ T)v=\mbox{ev}_{Tv}$

Woohoo! We just proved naturality. In fact, we did two things. First, we showed that we could embed $V$ in $V^{**}$, and we didn’t have to make any choices about elements to do so. This embedding is canonical. Second, we showed that linear transformations between $V$ and $W$ correspond to linear transformations between $V^{**}$ and $W^{**}$ in a nice way (such that the diagram commutes).