Jonathan Castello wrote:

>Can you elaborate on why we ought to consider \\(\mathbb{N}\\) as a category in this context? It seems we're only counting things, so a mere function \\(\mathrm{Ob}(C) \to \mathbb{N}\\) seems appropriate.

If we were just counting the object set of \\(C\\), that would be fine, but we're in fact counting possible *morphisms* of \\(C\\), ie the cardinality of \\(C(F(\text{Person}), F(\text{Person}))\\).

Since this operator is acting on morphisms as well as objects, it is a functor.

>Can you elaborate on why we ought to consider \\(\mathbb{N}\\) as a category in this context? It seems we're only counting things, so a mere function \\(\mathrm{Ob}(C) \to \mathbb{N}\\) seems appropriate.

If we were just counting the object set of \\(C\\), that would be fine, but we're in fact counting possible *morphisms* of \\(C\\), ie the cardinality of \\(C(F(\text{Person}), F(\text{Person}))\\).

Since this operator is acting on morphisms as well as objects, it is a functor.