ScalingStacks

Proof. Since $\qc(Y)\simeq\lim_{\it{Aff}/Y}\Mod_{B}$, the claim that $f_{+}M$ forms a quasi-coherent sheaf on $Y$ is equivalent to the claim that for any diagram of the form

$f_{+}M(g\circ h)$ is canonically equivalent to $h^{*}f_{+}M(g)$. Unraveling these formulas, by definition we have that $f_{+}M(g\circ h)=f^{\prime\prime}_{*}h^{\prime*}g^{\prime*}M$ and that $h^{*}f_{+}M(g)\simeq h^{*}f^{\prime}_{*}g^{\prime*}M$. By the previous lemma, these are equivalent by base change in the left hand square: $f^{\prime}:X\times_{Y}\Spec B\rightarrow\Spec B$ is perfect, and so $h^{*}f^{\prime}_{*}\simeq f^{\prime\prime}_{*}h^{\prime*}$. ∎