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.