The Push-Pull Formula for Quasicoherent Sheaves
Olivia Zamora 
This is Vakil 16.3 G, self-study.
Suppose $\psi: Z \to Y$ is any morphism, and both $\pi: X \to Y$ and $\pi': W \to Z$ are quasicompact and quasiseparated so that pushforwards send quasicoherent sheaves to quasicoherent sheaves. Suppose $\mathcal F$ is a quasicoherent sheaf on X. Suppose $\psi': W \to X$ commutes the diagram. Describe a natural morphism
$$\psi^* \pi_* \mathcal F \to \pi'_* \psi'^* \mathcal F$$
of quasicoherent sheaves on $Z$.
I started off assuming $X = \operatorname{Spec}A$, $Y = \operatorname{Spec}B$ since we know pullbacks can be taken affine-locally on the target. That would make $\mathcal F \simeq \tilde{M}$ for some $A$-module $M$, so that $\pi_* \mathcal F \simeq \tilde{M_B}$, where $M_B = M \otimes_A B$. Then, for any $\operatorname{Spec}C \subset Z$, we would have $\psi^* \pi_* \mathcal F \simeq \widetilde{M_B \otimes_B C}$ on $\operatorname{Spec} C$, which is naturally isomorphic to $\widetilde{M \otimes_A C}$. But when I run through the same computations using $\operatorname{Spec} D \subset W$ on the other side of the proposed morphism, I get the same result, so it seems as though I am getting an isomorphism, when I know this is incorrect from the following discussion in Vakil.
Where am I going wrong, and what is the most efficient correct argument?
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