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}

\\]

\\[

\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}

\\]