Dependent products and indexed left adjoints

First suppose that \(\phi^*\) admits a left adjoint \(\phi_{\sharp}\colon S\to T\). For every object \(X\in T\), the induced functor has a left adjoint which sends a map \(Y\to \phi^*(X)\) to its transpose \(\phi_{\sharp}(Y)\to X\).Now let \(p\colon X'\to X\) be a morphism in \(T\). Since \(\phi^*\) is left exact, we have a commutative square of pullback functors

The assertion that \(\phi^*\) preserves dependent products along \(p\) is precisely that the corresponding right Beck–Chevalley comparison is an isomorphism. Taking mates under the adjunctions \(\phi_{\sharp,X}\dashv \phi_X^*\), \(\phi_{\sharp,X'}\dashv \phi_{X'}^*\), \(p^*\dashv \Pi_p\), and \(\phi^*(p)^*\dashv \Pi_{\phi^*(p)}\), this is equivalent to the left Beck–Chevalley comparison being an isomorphism. Evaluating this comparison on a map \(Y\to \phi^*(X)\) gives the canonical map from the transpose of the pullback square

to the pullback of \(\phi_{\sharp}(Y)\to X\) along \(p\). Thus the left Beck–Chevalley comparison is an isomorphism exactly when every such transposed square is a pullback. This is precisely the condition that \(\phi_{\sharp}\) is \(T\)-indexed.It remains to note that preservation of dependent products implies that \(\phi^*\) has a left adjoint. Let \(\{X_i\}_{i\in I}\) be a small family of objects in \(T\). The product \(\prod_i X_i\) can be constructed as the dependent product of the object along the fold map \(\coprod_{i\in I} *\to *\). Since \(\phi^*\) preserves colimits, the coproducts in this construction are carried to the corresponding coproducts in \(S\), and by assumption \(\phi^*\) preserves the dependent product. Hence \(\phi^*\) preserves small products. Together with left exactness, this implies that \(\phi^*\) preserves all small limits. Since \(\phi^*\) is an accessible functor between presentable categories, the adjoint functor theorem gives a left adjoint to \(\phi^*\). The first part of the proof then shows that this left adjoint is \(T\)-indexed.