Is there a notion of “localization morphism” of schemes?

Let $$A$$ be a ring and $$S\subseteq A$$ a multiplicative system. Then the localization homomorphism of rings $$\phi:A\to S^{-1}A$$ induces a morphism between the spectra: $$f=\mathrm{Spec}(\phi):\mathrm{Spec}(S^{-1}A)\to\mathrm{Spec}(A)$$.

Is there a property of scheme morphisms $$f:X\to Y$$ that captures the idea that $$f$$ locally looks like the Spec of a localization homomorphism of rings?

Notice that $$S$$ doesn’t have to be of the form $$\{f^n\}_{n\geq 0}$$ for $$f\in A$$ or $$A\smallsetminus \mathfrak{p}$$ for a prime ideal $$\mathfrak{p}\subset A$$.

Here’s an attempt. $$f:X\to Y$$ is a localization morphism if there is an affine open cover $$\{V_i\}$$ of $$Y$$ such that every $$f^{-1}(V_i)$$ has an affine open cover $$\{U_i\}$$ such that the morphism $$f:U_i \to V_i$$ corresponds to a homomorphism $$\phi=f^{\sharp}:A\to B$$, where $$A=\mathcal{O}_Y(V_i)$$ and $$B=\mathcal{O}_X(U_i)$$, and there is a multuplicative system $$S\subseteq A$$ and an isomorphism $$\alpha:B\tilde{\to} S^{-1}A$$ such that $$\alpha\circ\phi=\lambda$$, where $$\lambda:A\to S^{-1}A$$ is the canonical map to the localization.