[002M] Construction 4.4.1·a (Closure of a class of sets under base change).

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 $\bar{\mathfrak{C}}$.