Jonathan wrote:

>Yes, but those are functors over posets, not morphisms _in_ a poset. You said "functors map functions", but monotone maps don't map functions, they map relationships \$$x \le y\$$ to relationships \$$f(x) \le f(y)\$$.

Monotone functions are \$$\mathbf{Bool}\$$-functors, not functors (\$$\mathbf{Set}\$$-functors). \$$\mathcal{V}\$$-functors are not necessarily functors, much the same way that \$$\mathcal{V}\$$-categories are not nessesarily categories.