One (Category) is Terminal Object

Theorem
Let $\mathbf{Cat}$ be the category of categories.

Let $\mathbf 1$ be the trivial category.

Then $\mathbf 1$ is a terminal object of $\mathbf{Cat}$.

Proof
Let $\mathbf C$ be an object of $\mathbf{Cat}$, i.e. a small category.

From Singleton is Terminal Object of Category of Sets, there exist unique mappings:


 * $F_0: \mathbf C_0 \to \mathbf 1_0 = \left\{{*}\right\}$
 * $F_1: \mathbf C_1 \to \mathbf 1_1 = \left\{{\operatorname{id}_*}\right\}$

since the latter sets are singletons.

It remains to verify that $F: \mathbf C \to \mathbf 1$ so defined, in fact is a functor.

Trivially, $F$ preserves identity morphisms.

That $F$ has the morphism property follows from:


 * $\operatorname{id}_* \circ \operatorname{id}_* = \operatorname{id}_*$

Hence $F: \mathbf C \to \mathbf 1$ constitutes a functor.

The result follows by definition of terminal object.