The Stacks project

Proof. It is clear that (1) implies (2) by considering $\text{id}_ X$. Assume $f : Y \to X$ is as in (2), and let $V \to Y$ be a surjective étale morphism from a scheme towards $Y$. Then the composition $V \to X$ is surjective étale as well and by Lemma 66.29.2 the pullback of $\mathcal{F}$ to $V$ is quasi-coherent as well. Hence we see that (2) implies (3).

Let $U \to X$ be as in (3). Let us use the abuse of notation introduced in Equation (66.26.1.1). As $\mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent there exists an étale covering $\{ U_ i \to U\} $ such that $\mathcal{F}|_{U_{i, {\acute{e}tale}}}$ has a global presentation, see Modules on Sites, Definition 18.17.1 and Lemma 18.23.3. Let $V \to X$ be an object of $X_{\acute{e}tale}$. Since $U \to X$ is surjective and étale, the family of maps $\{ U_ i \times _ X V \to V\} $ is an étale covering of $V$. Via the morphisms $U_ i \times _ X V \to U_ i$ we can restrict the global presentations of $\mathcal{F}|_{U_{i, {\acute{e}tale}}}$ to get a global presentation of $\mathcal{F}|_{(U_ i \times _ X V)_{\acute{e}tale}}$ Hence the sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ satisfies the condition of Modules on Sites, Definition 18.23.1 and hence is quasi-coherent.

The equivalence of (3) and (4) comes from the fact that any scheme has an affine open covering. $\square$