I am looking at Bart Jacob's book "Categorical Logic and Type Theory". The proof of Lemma 1.9.7 is left as an exercise for the reader. It does not seem that easy to me, and i have had quite limited success proving it. Here is the statement of the Lemma:
Consider a fibration for which each re-indexing functor has both a left $\coprod$ and right $\prod$ adjoint. The Beck-Chevalley holds for co-products $\coprod$ if and only if it holds for products $\prod$.
If i were able to prove the conjecture i would be able to prove the simpler conjecture that, given $\coprod \dashv \Delta \dashv \prod$, then the unit of $\coprod \dashv \Delta$ is iso if and only if the counit of $\Delta \dashv \prod$ is iso. This is a consequence of the lemma when you consider the split fibration where the base category is just an arrow, and that arrow has re-indexing functor $\Delta$. So i decided to practice the eventual proof by concentrating on that case first and then trying to generalise afterwards.
If $\eta$, the unit of $\coprod \dashv \Delta$, is iso, and $\pi$ is the counit of $\Delta \dashv \prod$ then looking at the triangle identities for the latter we quickly see that $\pi \Delta$ is a split epi, thus so is $\pi \Delta \coprod$ but since $\eta : 1 \cong \Delta \coprod$ we see $\pi$ too is a split epi, whence $\prod$ is faithful. To complete the simpler proof one needs to show $\prod$ full, $\pi$ a monic, or a number of other equivalent things, none of which i have been able to connect back to the hypothesis. Can anyone help me out with either the original conjecture or the simpler one?!
It's actually much easier than that. Recall that adjoints are unique up to unique isomorphism, and that the adjoint of a composite is the composite of the adjoints. So if we have a pullback square $$\begin{array}{ccc} \bullet & \stackrel{z}{\to} & \bullet \\ {\scriptstyle x} \downarrow & & \downarrow {\scriptstyle y} \\ \bullet & \stackrel{w}{\to} & \bullet \end{array}$$ and we have the Beck–Chevalley isomorphism $\Sigma_x z^* \cong w^* \Sigma_y $, then taking right adjoints of everything, we obtain the Beck–Chevalley isomorphism $y^* \Pi_w\cong \Pi_z x^*$.
Incidentally, given an adjunction $F \dashv G \dashv H$, it is true that the unit of $F \dashv G$ is a natural isomorphism if and only if the counit of $G \dashv H$ is a natural isomorphism: indeed, we have $G F \dashv G H$, so $G F$ is isomorphic to $\mathrm{id}$ if and only if $G H$ is isomorphic to $\mathrm{id}$.