So you're taking the family of maps $A \to \prod_{\mathfrak{p}} A/\mathfrak{p}$ and then interpret that product as a function of sets.
You can do the same thing with other families of maps, i.e. $A \to \prod_{\mathfrak{p}} A_{\mathfrak{p}}$ and $\prod_{\mathfrak{p}} A_{\mathfrak{p}} \to \prod_{\mathfrak{p}} \kappa(\mathfrak{p})$ and $\prod_{\mathfrak{p}} A/{\mathfrak{p}} \to \prod_{\mathfrak{p}} \kappa(\mathfrak{p})$
In general you have commutative diagram with $A$, $A/\mathfrak{p}$, $A_\mathfrak{p}$ and $\kappa(\mathfrak{p})$ and I suppose you can give interpretations to all those objects …