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.

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.