From Categories for the Working Mathematician pg. 43:
Theorem 1. The collection of all natural transformations is the set of arrows of two different categories under two different operations of composition, $\cdot$ and $\circ$, which satisfy the interchange law (5).
Question 1: What are these "two different categories"? The author never specifies.
Moreover, any arrow (transformation) which is an identity for the composition $\circ$ is also an identity for composition $\cdot$.
Question 2: What is an example of an identity for the composition $\circ$ also serving as an identity for the composition $\cdot$? And is this just an example where for $\tau, \sigma$ natural transformations then
$$ \tau \cdot I = \tau = I \cdot \tau \iff \tau \circ I = \tau = I \circ \tau $$
Note the objects for the horizontal composition $\circ$ are the categories, for the vertical composition, the functors.
Question 3: This doesn't make sense to me. Aren't the objects for horizontal composition the horizontal morphisms between categories (whose objects are categories)?
For example, if $C$ and $D$ are categories, then it seems to me the author is (nonsensically) saying that $C \circ D$ makes sense.