# Naturality (part 1)

I claim that a functor is really just a map on morphisms. In the definitions I gave, I said it was a map on categories in the sense that it was a map on morphisms and on objects. Well yes, but the part about how it affects objects is secondary in the following sense. Suppose I told you all the information about what a given functor $F$ does to morphisms. You could reconstruct what it does to objects by seeing “where the endpoints go.” If $f:A\to B$, then I could see what $F(A)$ is by considering the domain of $F(f)$.

This is to say, I can think of functors as maps on morphisms. This makes the following idea seem mildly reasonable:

• Morphisms are maps between objects (by definition)
• Functors are maps between morphisms (by the above argument)
• ??? are maps between functors

It almost seems “natural” to ask what’s next in the sequence. The answer is “natural transformations.” That is, we define a natural transformation to be what comes next.

So technically, here’s the definition. Let $F,G$ be functors between categories $\mathcal C$ and $\mathcal D$. Then a natural transformation is a collection of maps $\eta=\{\eta_X:F(X)\to G(X)\mid X\in Ob(\mathcal C)\}$ such that, for any map $f:X\to Y$, the following diagram commutes:

Let’s unravel this. If I input a functor $F$ into a natural transformation $\eta$, it should spit out another functor $G$. Functors are maps on morphisms, so I need to tell you what $G=\eta(F)$ does to morphisms. That is, I need to tell you what $G(f)$ is. This diagram tells you how to get $G(f)$ from $\eta$ and $F(f)$.

Yes, I get it, you are probably confused. This sort of thing happens to all of us in category theory. But if you push through it, you’ll at least get used to how it works, even if you don’t feel like you’re gaining understanding.