I want to prove Theorem 1 in section 2.5 of McLane's Categories for the Working Mathematician, which was left to the reader.
The theorem:
The collection of all natural transformations is the set of arrows of two different categories under different operations of composition; $\circ$ and $*$ (using the notation I'm used to for vertical and horizontal respectively) which satisfy the interchange law. Moreover, any arrow (natural transformation) which is an identity for $*$ is also an identity for $\circ$.
If I am reading this correctly, I need to show that every natural transformation is an arrow in both of these categories.
There is a hint that for the horizontal composition $*$, the category is categories as its objects, while for the vertical composition $\circ$ the category has functors as its objects.
The latter case is simple, since the morphisms in a functor category are natural transformations, and this is a category under vertical composition. For the first category then, I'll take the category whose objects are all functors (between all categories). Then every natural transformation is an arrow in this category and every arrow in this category is a natural transformation.
What is the second category? Below is my attempt, but I'm not sure I'm on the right track.
For the former case, I note that any functor $F: A \rightarrow B$ induces a natural transformation $\overline{F}: hom_A \rightarrow hom_B \circ (F \times F)$ by its arrow functions $\overline{F}_{a, b} = F_{a, b} = f \mapsto Ff : hom_A(a, b) \rightarrow hom_B(Fa, Fb)$.
So considering the category of categories as the second category, with arrows (functors) interpreted as natural transformations per above. Composition of functors induces a composition of the induced natural transformations by (for which I'll repurpose $*$, noting this doesn't seem to be the same as horizontal composition) $(\overline{G} * \overline{F})_{a, b} = (\overline{GF})_{a, b} = \overline{G}_{Fa, Fb} \circ \overline{F}_{a, b}$.
I dont think that I have this second category correct, since I think I am supposed to interpret natural transformation as arrows in a category in which composition induces the horizontal composition of natural transformations, but I can't see what that category would be.