Infimum is Product in Order Category

Theorem
Let $\mathbf P$ be an order category whose ordering is $\preceq$.

Let $p,q \in \mathbf P_0$, and suppose that they have some infimum $r = \inf \left\{{p, q}\right\}$.

Then $r$ is a binary product of $p$ and $q$ in $\mathbf P$.

Proof
Suppose that there are morphisms $l \to p$ and $l \to q$ in $\mathbf P$.

That is to say, suppose $l \preceq p$ and $l \preceq q$.

Then $l$ is a lower bound for $\left\{{p, q}\right\}$.

By definition of infimum, we then have:


 * $l \preceq r = \inf \left\{{p, q}\right\}$

By definition of $\mathbf P$ as an order category, this means there is a morphism:


 * $l \to r$

By definition of $\mathbf P$, this morphism is necessarily unique.

Thus $r$ satisfies the UMP for the binary product of $p$ and $q$.

The result follows.

Also see

 * Supremum is Coproduct in Poset Category, the dual result