Lemma 84.28.3. Let $f : V \to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^ n_ j : U_ n \to U_{n - 1}$ are flat and the morphisms $V_ n \to U_ n$ are quasi-compact and quasi-separated. Then $f^*$ and $f_*$ form an adjoint pair of functors between the categories of quasi-coherent modules on $U_{Zar}$ and $V_{Zar}$.

Proof. We have seen in Lemmas 84.28.1 and 84.28.2 that the statement makes sense. The adjointness property follows immediately from the fact that each $f_ n^*$ is adjoint to $f_{n, *}$. $\square$

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