## § Try and think of natural transformations as intertwinings

I'm comfrotable with elementary representation theory, but I feel far less at home manipulating
natural transformations. I should try and simply think of them as the intertwinig operators
in representation theory, since they do have the same diagram. Then the functors become
two representations of the same category (group), and the natural transformation is an
intertwining operator.
If one does this, then Yoneda sort of begins to look like Schur's lemma. Schur's
lemma tells us that intertwinings between irreducible representations are either zero
or a scaling of the identity matrix. That is, they are one-dimensional, and the space
of all intertwinings is morally isomorphic to the field $\mathbb C$. If we specialize
to character theory of cyclic groups $Z/nZ$, let's pick one representation to be
the "standard representation" $\sigma: x \mapsto e^{i 2 \pi x/n}$. Then, given some other
representation $\rho: Z/nZ \rightarrow \mathbb C^\times$, the intertwining between
$\sigma$ and $\rho$ is determined by where $\rho$ sends $1$. If $\rho(1) = k \sigma(1)$
for $k \in \mathbb R$, then the intertwining is scaling by $k$. Otherwise, the intertwining
is zero. This is quite a lot like Yoneda, where the natural transformation is fixed
by wherever the functor sends the identity element.