**Proof.**
Assume $\mathcal{F}$ is quasi-coherent. Then $\mathcal{F}$ is a sheaf for the fppf topology, hence a sheaf for the étale topology. Moreover, any pullback of $\mathcal{F}$ to a ringed topos is quasi-coherent, hence the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ are quasi-coherent. This proves $\mathcal{F}$ is locally quasi-coherent. Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. We have seen that $\mathcal{X}/y = (\mathit{Sch}/V)_{fppf}$. By Descent, Proposition 35.8.9 it follows that $y^*\mathcal{F}$ is the quasi-coherent module associated to a (usual) quasi-coherent module $\mathcal{F}_ V$ on the scheme $V$. Hence certainly the comparison maps (96.9.4.1) are isomorphisms.

Conversely, suppose that $\mathcal{F}$ satisfies (1) and (2). Let $y$ be an object of $\mathcal{X}$ with $V = p(y)$. Denote $\mathcal{F}_ V$ the quasi-coherent module on the scheme $V$ corresponding to the restriction $y^*\mathcal{F}|_{V_{\acute{e}tale}}$ which is quasi-coherent by assumption (1), see Descent, Proposition 35.8.9. Condition (2) now signifies that the restrictions $x^*\mathcal{F}|_{U_{\acute{e}tale}}$ for $x$ over $y$ are each isomorphic to the (étale sheaf associated to the) pullback of $\mathcal{F}_ V$ via the corresponding morphism of schemes $U \to V$. Hence $y^*\mathcal{F}$ is the sheaf on $(\mathit{Sch}/V)_{fppf}$ associated to $\mathcal{F}_ V$. Hence it is quasi-coherent (by Descent, Proposition 35.8.9 again) and we see that $\mathcal{F}$ is quasi-coherent on $\mathcal{X}$ by Lemma 96.11.3.
$\square$

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