Singleton is Terminal Object of Category of Sets

Theorem
Let $\mathbf{Set}$ be the category of sets.

Let $S = \left\{{x}\right\}$ be any singleton set.

Then $S$ is a terminal object of $\mathbf{Set}$.

Proof
Let $T$ be a set, and let $f: T \to S$ be a mapping.

Then since for all $t \in T$, we have $f \left({t}\right) \in S$, it follows that:


 * $\forall t \in T: f \left({t}\right) = x$

By Equality of Mappings, there is precisely one such mapping $f: T \to S$.

Hence the result, by definition of terminal object.