Well the category \$$\mathbf{2}\$$,

\$\ast \rightarrow \bullet, \$

is really, up to isomorphism (aka relabeling), just our old friend \$$\mathbf{Bool}\$$,

\$\texttt{false} \rightarrow \texttt{true}, \$

which, because if such functors into \$$\mathbb{N}\$$ out of \$$\mathbf{Set}\$$ must factor through \$$\mathbf{2} \cong \mathbf{Bool} \$$, that would imply that such "functors" are exactly the *monotone functions* we studied in the first two chapters.

Edit, I believe Simon's constraints must amount to the diagram,

\$\begin{matrix} \mathbf{Set }& & \\\\ f \downarrow & \overset F \searrow & \\\\ \mathbf{2} & \begin{matrix} \overset l \leftarrow \\\\ \underset r\rightarrow \end{matrix} & \mathbb{N}. \end{matrix} \$