Categories for which Natural Transformations are the Set of Arrows 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
$$
\tau \cdot I = \tau = I \cdot \tau \iff \tau \circ I = \tau = I \circ \tau
$$
holds?

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.

The horizontal category is:


*

*Objects are categories

*Morphisms are natural transformations

*The product of morphisms is horizontal composition $\circ$

*The identity for an object $C$ is $1_{1_{\mathcal{C}}}$


The vertical categtory is


*

*Objects are functors

*Morphisms are natural trasnformations

*The product of morphisms is vertical composition $\cdot$

*The identity for an obect $F$ is $1_F$


And note that in the vertical category, if $C$ is a category, the identity for the object $1_\mathcal{C}$ is $1_{1_{\mathcal{C}}}$

Incidentally, a category is determined (up to isomorphism) by its set of morphisms and composition law; there are even "arrow only" axiomatizations of categories (basically, you replace the notion of "object" with that of "identity arrow"). So, the text of theorem 1 actually does specify what the two categories are.
