If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the single element of S. Thus every singleton is a terminal object in the category of sets.
A singleton has the property that every function from it to any arbitrary set is injective. The only non-singleton set with this property is the empty set.
Then S is called a singleton if and only if there is some y ∈ X such that for all x ∈ X,
That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)