## § 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.