 Solution

A poset $P$ becomes a category with the elements of $P$ as objects and exactly one morphism $x\to y$ when $x\le y$, with no morphism otherwise. Reflexivity supplies the identity morphisms and transitivity supplies composition in a category; uniqueness of morphisms proves associativity. A functor $f:P\to Q$ is exactly a monotone function: it must send $x\le y$ to $f(x)\le f(y)$, and this condition suffices because all the required morphisms are unique. Antisymmetry says that isomorphic objects of this category are equal.