Let $\mathfrak{C}\subseteq\mathscr{M}$ be a class of sets; there exists a class $\bar{\mathfrak{C}}$ of *families*
of sets that contains $\prn{S\Sub{i}}\Sub{i\in I}$ exactly when each
$S\Sub{i}$ lies in $\mathfrak{C}$. We will refer to $\bar{\mathfrak{C}}$ as the *closure under base change* of $\mathfrak{C}$, a name motivated by the fact that when $\prn{S\Sub{i}}\Sub{i\in I}$ lies in $\bar{\mathfrak{C}}$,
then for any $u:J\to I$, the base change $\prn{S\Sub{uj}}\Sub{j\in J}$ also lies in $\bar{\mathfrak{C}}$.