Infimum is Product in Order Category

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

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

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

Also see

 * Supremum is Coproduct in Poset Category, the dual result