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}\\)?

\\[

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}\\)?