 Solution

A poset becomes a category whose objects are its elements and whose hom-set $X(x,y)$ is a singleton if $x\le y$ and empty otherwise. Reflexivity supplies identity arrows and transitivity supplies composition; all possible composites are unique, so the category axioms hold. Antisymmetry says that isomorphic objects are equal. A functor between such categories is exactly an order-preserving function: an inequality must be sent to an inequality, and there is no further choice of arrow map.