ScalingStacks

Proof. Since $g$ is affine and hence $g^{\prime}$ is as well, the fiber product $X\times_{A}\Spec B$ can be identified with the relative spectrum $\Spec_{X}C$ of a commutative algebra object $C\in{\rm Alg}(\qc(X))$, and $g^{\prime}_{*}$ induces an equivalence $\qc({X\times_{A}\Spec B})\simeq\Mod_{C}(\qc(X))$. Furthermore, the pullback $g^{\prime*}$ can be described as tensoring with $C$, and thus in particular $f^{\prime}_{*}g^{\prime*}M\simeq f_{*}(C\otimes M)$. However, the global sections functor takes fiber products to tensor products, so we can identify $C\simeq f^{*}g_{*}B$. Applying the previously established projection formula twice, we can now compute $f_{*}(f^{*}g_{*}B\otimes M)\simeq f_{*}M\otimes_{A}g_{*}B\simeq g^{*}f_{*}M\otimes_{B}B\simeq g^{*}f_{*}M$, completing the proof. ∎