Let me try again. I was trying to make a statement by asking a question, but it seems that some ambiguity in notion of generalizing a singleton led to confusion. To rephrase the question:

Within the category of Set, how can we abstractly characterize a singleton set -- meaning, without looking inside of it, just looking at the arrows in which it participates?

Well, if you picture the category Set as a directed graph, with one node for each set, and a directed edge \$$f: A \rightarrow B\$$ for every function from A to B, the answer is easy to see. (This is a multigraph, because there can be self-loops and multiple arrows from A to B.)

What's special about a singleton set Y in this graph? For any set Y (including X itself), there is one and only one edge going from X to Y. Since Y contains only one member, there is only one way to construct a function from X to Y. This is called a **terminal object** in the graph.

And singletons are the only terminal objects in the graph.

So to saying that a set is a terminal object is logically equivalent to saying that it contains a single member.

It is the abstract, categorical way of saying it, because it only involves the relationships, and makes no reference to any internal structures of the object.