Does the comma category of a set over category of graphs have initial and terminal objects?

Question

As the question states. Let $S$ be a set and $U : \textbf{Graph} \to \textbf{Set}$ be the forgetful functor. Then does the comma category $S \downarrow U$ have initial or terminal objects?

I think the answer is no to both questions. Every construction of an initial/terminal object that I've tried has failed for some reason or another. Maybe I'm thinking about the question in the wrong way.

Answer 1

An object of $S \downarrow U$ is an $S$-pointed graph, so there definitely is an initial object: which $S$-pointed graph has a unique $S$-embedding into every graph?

Answer: $S$ as a discrete graph with the natural $S$-pointing.

If you allow loops, there also is a terminal object: which $S$-pointed graph admits a unique $S$-homomorphism from any graph?

Answer: $S$ as a complete graph with all edges, including all loops.