Supppose that $\displaystyle \{ X_ \alpha \} _ { \alpha \in A } $ is an indexed family of sets, and define their Cartesian product $\displaystyle \pi_ { \alpha \in A } X_ \alpha $ to be the set of all maps $\displaystyle f: A \rightarrow \bigcup _{ \alpha \in A } $ such that $\displaystyle f( \alpha ) \in X_ \alpha \ \ \ \ \ \forall \alpha \in A $

Find the inverse function of the Cartestian product, $\displaystyle \pi ^ {-1} _ { \alpha } (E) $ for $\displaystyle E _ \alpha \subset Z_ \alpha $

My solution:

The inverse function of the cartestian product should be all the inverse maps $\displaystyle f^{-1} : E \rightarrow Z_ \alpha $ which is defined by $\displaystyle f^{-1} = \{ \alpha \in A : f( \alpha ) \in E _ { \alpha } \subset Z _ \alpha \}$

But is that the best we can do? Thanks.