Coproduct of Ordered Sets

Theorem
Let $\mathbf{Pos}$ be the category of posets.

Let $\left({P, \preceq_1}\right)$ and $\left({Q, \preceq_2}\right)$ be posets.

Let $P \sqcup Q$ be the disjoint union of $P$ and $Q$.

Let $\preceq$ be the ordering on $P \sqcup Q$ defined by:


 * $\left({x, i}\right) \preceq \left({y, j}\right)$ iff $i = j$ and $x \preceq_i y$

where $i = 1$ or $i = 2$ depending on whether $x,y \in P$ or $x,y \in Q$.

Then $\left({P \sqcup Q, \preceq}\right)$ is the coproduct of $P$ and $Q$ in $\mathbf{Pos}$.