"Mild problem" - like it completely doesn't work? \$$\qquad \$$

Simon is much nicer than I am.

So is this a reasonable revised conjecture?

**Revised Conjecture.** Every functor \$$F: \mathbf{Set} \to \mathbb{N}\$$ is of this form: \$$F\$$ sends every object to \$$\star\$$, it sends every morphism \$$f: \emptyset \to Y\$$ to the same morphism \$$n : \star \to \star\$$, and it sends every morphism \$$f: X \to Y\$$ with \$$X \ne \emptyset\$$ to the identity morphism \$$1\_\star : \star \to \star\$$.

Perhaps it's better to phrase this conjecture in terms of functors from \$$\mathbf{Set}\$$ to the category Simon brought in, also known as "\$$\mathbf{2}\$$": the category with two objects and one morphism from the first to the second.