Definition 4.49.
Let \(\phi^*\colon T \to S\) be a morphism of logoi. We say that \(\phi^*\) preserves dependent products if, for every morphism \(p\colon X \to Y\) in \(T\), the canonical comparison
\[\phi^*(\Pi_p Z) \longrightarrow \Pi_{\phi^*(p)}\phi^*(Z)\]
is an isomorphism for every \(Z \in T_{/X}\).