Indexed left adjoints

Let \(\phi^*\colon T \to S\) be a morphism of logoi and suppose that \(\phi^*\) admits a left adjoint \(\phi_{\sharp}\colon S \to T\). We say that \(\phi_{\sharp}\) is \(T\)-indexed if, for every morphism \(p\colon X' \to X\) in \(T\) and every pullback square in \(S\) of the form

the transposed square

is a pullback in \(T\).