A power is a limit over what diagram?

Question

A product in a category $\mathcal{C}$ is a limit over a diagram $F:S\to \mathcal{C}$ where $S$ is a set and is to be viewed as a discrete category.

I am struggling to see how a power $x^k$ of an object $x\in \mathcal{C}$ is to be defined in this way. Do we take $S$ to have $k$ identical objects? But then isn't $S$ just a one-object category?

Thanks!

Answer 1

The category $S$ is a discrete category with $k$ different objects. The functor $F$ then sends all of these objects to the same object $x$ of $\mathcal{C}$. So the objects of $S$ aren't identical, they just get identified by $F$.