Wait, since a set/finite set is a discrete preordering, and if we treat \$$\mathbb(N)\$$ as a preorder, could we define not a functor (\$$\mathbf{Set}\$$-functor), but rather a monotone function (\$$\mathbf{Bool}\$$-functor) (in fact, a strict monoidal monotone)?

\$F\lbrace \varnothing \rbrace = 0, \\\\ F(A \cup B) =F(A) +F(B) \$

Edit: Also, let's not forget,

\$A \subseteq B \Leftrightarrow F(A) \leq F(B) \$

Is \$$\mathbf{Bool}\$$ in some way "a level down" from \$$\mathbf{Set}\$$?