Matthew wrote:

> I argue this conjecture is _false_.

Hey, I think you're right! If I get the gist of your argument, it goes like this. Pick a nonempty set \$$S\$$ and define a functor \$$F: \mathbf{Set} \to \mathbf{N}\$$ that sends the unique function \$$f: \emptyset \to S\$$ to some non-identity morphism in \$$\mathbf{N}\$$ and all other morphisms to the identity. Check that this is a functor.

(We need \$$S\$$ to be nonempty so that \$$f\$$ isn't an identity morphism, so that \$$F(f)\$$ can be chosen to be a non-identity morphism.)

Very nice!