Smallest Element is Initial Object

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

Suppose $\mathbf P_0$, considered as a poset, has a smallest element $p$.

Then $p$ is an initial object of $\mathbf P$.

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


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

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

That is, $p$ is initial.