Natural Transformations Without Objects

Question

So I've been thinking about the definition of categories as just arrows with a defined composition (i.e. without objects). I understand this is silly, but it's fun and I have a question about it: functors are easy to define in this setting; they're just maps that preserve the composition. But how does one discuss natural transformations in this setting, since we don't have objects around to "index" them by? I suppose we could index by identity arrows, but this feels wrong somehow, and I'm sure one of you smart people has a better answer.

Answer 1

A natural transformation of functors $\mathbb{C} \to \mathbb{D}$ is the same thing as a functor $\mathbb{2} \times \mathbb{C} \to \mathbb{D}$, where $\mathbb{2} = \{ 0 \to 1 \}$. The domain of the natural transformation is the restriction to $\{ 0 \} \times \mathbb{C}$, and the codomain is the restriction to $\{ 1 \} \times \mathbb{D}$.

Answer 2

Identity arrows are it. For an identity $x$ and functors $F,G$, the component of a natural transformation $\sigma:F\to G$ at $x$ is going to be a morphism for which $F(x)$ and $G(x)$ are left and right identities, respectively.

Keep in mind in the normal definition of a category, there's a bijection between objects and their identities; indexing with either one gives you essentially the same natural transformation even before we consider tossing out the objects.