> 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.