The Stacks project

Proof. We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$. We claim that $f : \mathcal{X} \to \mathcal{Y}$ is a special cocontinuous functor, see Sites, Definition 7.29.2 which will prove the lemma. By Stacks, Lemma 8.10.3 the functor $f$ is continuous and cocontinuous. By Stacks, Lemma 8.8.1 we see that conditions (3), (4), and (5) of Sites, Lemma 7.29.1 hold. $\square$

Proof. We may assume $\mathcal{Y}$ is the stackification of $\mathcal{X}$. The first assertion is clear from Lemma 96.13.1 and $\mathcal{O}_\mathcal {X} = f^{-1}\mathcal{O}_\mathcal {Y}$. Pullback of quasi-coherent sheaves are quasi-coherent, see Lemma 96.11.2. Hence it suffices to show that if $f^*\mathcal{G}$ is quasi-coherent, then $\mathcal{G}$ is. To see this, let $y$ be an object of $\mathcal{Y}$. Translating the condition that $\mathcal{Y}$ is the stackification of $\mathcal{X}$ we see there exists an fppf covering $\{ y_ i \to y\}$ in $\mathcal{Y}$ such that $y_ i \cong f(x_ i)$ for some $x_ i$ object of $\mathcal{X}$. Say $x_ i$ and $y_ i$ lie over the scheme $U_ i$. Then $f^*\mathcal{G}$ being quasi-coherent, means that $x_ i^*f^*\mathcal{G}$ is quasi-coherent. Since $x_ i^*f^*\mathcal{G}$ is isomorphic to $y_ i^*\mathcal{G}$ (as sheaves on $(\mathit{Sch}/U_ i)_{fppf}$ we see that $y_ i^*\mathcal{G}$ is quasi-coherent. It follows from Modules on Sites, Lemma 18.23.3 that the restriction of $\mathcal{G}$ to $\mathcal{Y}/y$ is quasi-coherent. Hence $\mathcal{G}$ is quasi-coherent by Lemma 96.11.3. $\square$