Suppose we have a split fibration $p : \mathbb{E}\to\mathbb{B}$ with (split) simple products. To fix notation, this means that for every projection $\pi_{I,J} : I\times J \to I$ in the base category, the reindexing functor $\pi_{I,J}^* : \mathbb{E}_{I} \to \mathbb{E}_{I\times J}$ has a right adjoint $\forall_{I,J}$ and these functors satisfy the Beck-Chevalley condition, meaning that the canonical natural transformation $u^*\circ\forall_{K,J} \Rightarrow \forall_{I,J}\circ (u\times \text{id})^*$ is identity for every $u : I\to K$ in $\mathbb{B}$. This canonical morphism is the transpose of $(u\times \text{id})^*(\varepsilon_X)$.
My question is, is this enough for the pair $u^*$ and $(u\times \text{id})^*$ to be a map of adjunctions. Actually, it would be enough to show that reindexing functors preserve units of the adjunctions, meaning $u^*(\eta_X) = \eta'_{u^*(X)}$ for each $X$ and where $\eta$ is the unit of $\pi_{K,J}^* \dashv \forall_{K,J}$ and $\eta'$ of $\pi_{I,J}^* \dashv \forall_{I,J}$.
It seems to me that this really ought to hold and the proof should use the fact that the "canonical" morphism is the identity, but I couldn't produce one.