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.

>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.