# Order isomorphic implies order preserving if and only if maps are identity maps

Show that the two ordered sets $(P,\leq)$ and $(Q,\leq)$ are order isomorphic if and only if there exist order preserving maps $$X: P\to Q \\ Y: Q \to P$$ such that $X\circ Y = \text{id}_Q$ and $Y\circ X = \text{id}_P$, $\text{id}$ being identify map.

Where and how do we use the identity map ?