Let $I$ be an arbitrary index set and $A_i$ $(i\in I)$ a family of sets.
Is there a commonly used shorthand notation to denote the set $$S =\{(i, a) \mid i \in I, a \in A_i\}?$$
A related construct is the Cartesian product $\prod_{i \in I}A_i$, but each element of this set is a function $f: I \to \cup_{i\in I} A_i$ such that $f(i) \in A_i$ for each $i \in I$ (so that the graph of $f$ is a subset of $S$).