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?)