Singleton is Terminal Object of Category of Sets

Theorem

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

Let $S = \set x$ 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 $\map f t \in S$, it follows that:

$\forall t \in T: \map f t = x$

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

Hence the result, by definition of terminal object.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $1$