Show that when \(P\) is a discrete preorder, then every function \(P \rightarrow Q\) is a monotone map, regardless of the order \(\leq_Q\).
The only time the monotonicity criterion is deployed is when two elements of \(P\) are related, and for a discrete category this means we only have to check whether \(f(a) \leq_Q f(a)\), which is true because preorders are reflexive.