Suppose that \(\mathfrak {C}\) is a definable class of sets. To
check that \(\bar {\mathfrak {C}}\) is a definable class of families of sets,
we fix a family \({\mathopen {}\left (S_{i}\right )\mathclose {}}_{i\in I}\) not necessarily lying in \(\bar {\mathfrak {C}}\). Because \(\mathfrak {C}\) is definable, the intersection \(\mathfrak {C}\cap \bigcup _{i\in I}S_{i}\) is represented by a set \(U\). We therefore take the subset \(J = {\mathopen {}\left \{i \in I\mid S_i\in U\right \}\mathclose {}}\subseteq I\), and verify that the base change \({\mathopen {}\left (S_{j}\right )\mathclose {}}_{j\in J}\) is the largest approximation of \({\mathopen {}\left (S_{i}\right )\mathclose {}}_{i\in I}\) by a family lying in \(\bar {\mathfrak {C}}\).
Conversely suppose that \(\bar {\mathfrak {C}}\) is a definable class of families of sets. To see that \(\mathfrak {C}\) is definable, we fix a class \(\mathfrak {U}\) represented by a set \(U\in \mathscr {M}\) to check that \(\mathfrak {C}\cap \mathfrak {U}\) is representable. We consider the family of sets \({\mathopen {}\left (u\right )\mathclose {}}_{u\in U}\); because \(\bar {\mathfrak {C}}\) is definable, there is a largest subset \(V\subseteq U\) such that the change of base \({\mathopen {}\left (v\right )\mathclose {}}_{v\in V}\) lies in \(\bar {\mathfrak {C}}\), i.e. such that each \(v\in V\) lies in \(\mathfrak {C}\). Therefore \(\mathfrak {C}\cap \mathfrak {U}\) is represented by the set \(V\).