How do we define morphisms on a functor?

Question

Let $F:C\to Set$ be a functor. Let $\int F$ represent the category of elements. We shall define a functor $\int(-):Set^C\to Cat/C$ where $\int(-)(F):=\int F$. Now let $G,H:C\to Set$ be functors and $\alpha:G\Rightarrow H$ be a natural transformation. My question is how do we define $\int(-)(\alpha)$?

Answer 1

As usual in category theory, there's only really one definition that makes sense, and that definition is the one you want.

In particular, given a natural transformation $\alpha : F \to G$, you need to define a functor $$\textstyle \int \alpha : \int F \to \int G$$ Given an object $(C,x)$ of $\int F$, note that $(C, \alpha_C(x))$ is an object of $\int G$, so this suggests defining $(\int \alpha)(C,x) = \alpha_C(x)$ as the action of $\int \alpha$ on objects.

See if you can figure out how to define $\int \alpha$ on morphisms.