Smallest Element is Initial Object

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Theorem

Let $\mathbf P$ be an order category.

Suppose the objects $\mathbf P_0$ of $\mathbf P$, considered as an ordered set, have 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:

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

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


That is, $p$ is initial.

$\blacksquare$


Sources