# Greatest Element is Terminal Object

## Theorem

Let $\mathbf P$ be an order category.

Let $p$ be the greatest element of the objects $\mathbf P_0$ of $\mathbf P$, considered as a ordered set.

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

## Proof

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

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

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

That is, $p$ is terminal.

$\blacksquare$