Greatest Element is Terminal Object

Theorem
Let $\mathbf P$ be a poset category.

Let $p$ be the greatest element of $\mathbf P_0$ considered as a poset.

Then $p$ is a terminal object of $\mathbf P$.

Proof
Since $p$ is the greatest element of $\mathbf P_0$, we have:


 * $\forall q \in \mathbf P_0: q \le p$

i.e., for every object $q$ of $\mathbf P$ there is a unique morphism $q \to p$.

That is, $p$ is terminal.