Supremum is Coproduct in Order Category

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

Let $p, q \in P_0$, and suppose they have some supremum $r = \sup \, \left\{{p, q}\right\}$.

Then $r$ is the coproduct of $p$ and $q$ in $\mathbf P$.

Also see

 * Infimum is Product in Poset Category, the dual result