Polynomial overrings of $\mathrm{I}\mathrm{n}\mathrm{t}(\mathbb{Z})$

We show that every polynomial overring of the ring $\mathrm{I}\mathrm{n}\mathrm{t}(\mathbb{Z})$ of polynomials which are integer-valued over $\mathbb{Z}$ may be considered as the ring of polynomials which are integer-valued over some subset of $\widehat{\mathbb{Z}}$, the profinite completion of $\mathbb{Z}$ with respect to the fundamental system of neighbourhoods of $0$ consisting of all non-zero ideals of $\mathbb{Z}\mathrm{.}$