I think the above result comes more or less immediately when you consider that a finitely supported function \\(S \to \mathbb{N}\\) is no more than \\(\mathbb{N}^S\\), the \\(S\\)-indexed product on \\(\\mathbb{N}\\). Restriction discards some of these copies of \\(\mathbb{N}\\); to recover them, we need either the minimal or maximal element of \\(\mathbb{N}\\), depending on if we want a left adjoint or a right adjoint. \\(\mathbb{N}\\) has one, but not the other.

(I think I just invoked the fact that \\(Set\\) is Cartesian closed. Is that the case?)

(I think I just invoked the fact that \\(Set\\) is Cartesian closed. Is that the case?)