[002P] Definition 4.4.1·c (Definable class of families of sets).

A class of families of sets $\mathfrak{F}$ is said to be definable when it is stable and moreover, for any family of sets $\prn{S\Sub{i}}\Sub{i\in I}$, there exists a subset $J\subseteq I$ such that the base change $\prn{S\Sub{j}}\Sub{j\in J}$ lies in $\mathfrak{F}$, and moreover, such that $u:K\to I$ factors through $J\subseteq I$ whenever the base change $\prn{S\Sub{uk}}\Sub{k\in K}$ lies in $\mathfrak{F}$.