Monotone maps, considering the source and target preorders as Bool-categories, are in fact Bool-functors.
The monotone map constraint, that \(x_1\ \leq_X\ x_2 \implies F(x_1)\leq_Y F(x_2)\), translates to the enriched category functor constraint, that \(\mathcal{X}(x_1,x_2) \leq \mathcal{Y}(F(x_1),F(x_2))\).