For any preorder, the identity function is a monotone map.
The composition of two monotone maps (\(P \xrightarrow{f} Q \xrightarrow{g} R\)) is also monotone.
Monotonicity translates to \(a \leq b \implies a \leq b\) and is trivially true.
Need to show: \(a \leq_P b \implies g(f(a)) \leq_R g(f(b))\)
The monotonicity of \(f\) gives us \(f(a) \leq_Q f(b)\) and the monotonicity of \(g\) gives us the result we need.