> Each functor corresponds to a morphism and vice versa.

But you've stated that the objects of \$$\mathbf{Mult}\$$ are the morphisms of \$$\mathbf{N}\$$. Is this not a something of a level slip?

I would have expected \$$\textbf{Mult}\$$ to be the single-object category with \$$\mathbf{N}\$$ as its object, and functors \$$\mathbf{N} \to \mathbf{N}\$$ as morphisms. After all, the set of endofunctors on \$$\mathbf{N}\$$ forms a commutative monoid by their identification with integer scaling, and monoids are the same as single-object categories.