Proof.
In the arguments below x denotes an arbitrary object of \mathcal{X} lying over the scheme U. To show that an object \mathcal{H} of \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) is in \textit{LQCoh}(\mathcal{O}_\mathcal {X}) we will show that the restriction x^*\mathcal{H}|_{U_{\acute{e}tale}} = \mathcal{H}|_{U_{\acute{e}tale}} is a quasi-coherent object of \textit{Mod}(U_{\acute{e}tale}, \mathcal{O}_ U).
Proof of (1). Let \mathcal{I} \to \textit{LQCoh}(\mathcal{O}_\mathcal {X}), i \mapsto \mathcal{F}_ i be a diagram. Consider the object \mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i of \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}). The pullback functor x^* commutes with all colimits as it is a left adjoint. Hence x^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i. Similarly we have x^*\mathcal{F}|_{U_{\acute{e}tale}} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i|_{U_{\acute{e}tale}}. Now by assumption each x^*\mathcal{F}_ i|_{U_{\acute{e}tale}} is quasi-coherent. Hence \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i|_{U_{\acute{e}tale}} is quasi-coherent by Descent, Lemma 35.10.3. Thus x^*\mathcal{F}|_{U_{\acute{e}tale}} is quasi-coherent as desired.
Proof of (2). It follows from (1) that cokernels exist in \textit{LQCoh}(\mathcal{O}_\mathcal {X}) and agree with the cokernels computed in \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}). Let \varphi : \mathcal{F} \to \mathcal{G} be a morphism of \textit{LQCoh}(\mathcal{O}_\mathcal {X}) and let \mathcal{K} = \mathop{\mathrm{Ker}}(\varphi ) computed in \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}). If we can show that \mathcal{K} is a locally quasi-coherent module, then the proof of (2) is complete. To see this, note that kernels are computed in the category of presheaves (no sheafification necessary). Hence \mathcal{K}|_{U_{\acute{e}tale}} is the kernel of the map \mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{G}|_{U_{\acute{e}tale}}, i.e., is the kernel of a map of quasi-coherent sheaves on U_{\acute{e}tale} whence quasi-coherent by Descent, Lemma 35.10.3. This proves (2).
Proof of (3). Let 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 be a short exact sequence of \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}). Since we are using the étale topology, the restriction 0 \to \mathcal{F}_1|_{U_{\acute{e}tale}} \to \mathcal{F}_2|_{U_{\acute{e}tale}} \to \mathcal{F}_3|_{U_{\acute{e}tale}} \to 0 is a short exact sequence too. Hence (3) follows from the corresponding statement in Descent, Lemma 35.10.3.
Proof of (4). Let \mathcal{F} and \mathcal{G} be in \textit{LQCoh}(\mathcal{O}_\mathcal {X}). Since restriction to U_{\acute{e}tale} is given by pullback along the morphism of ringed topoi U_{\acute{e}tale}\to (\mathit{Sch}/U)_{\acute{e}tale}\to \mathcal{X}_{\acute{e}tale} we see that the restriction of the tensor product \mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G} to U_{\acute{e}tale} is equal to \mathcal{F}|_{U_{\acute{e}tale}} \otimes _{\mathcal{O}_ U} \mathcal{G}|_{U_{\acute{e}tale}}, see Modules on Sites, Lemma 18.26.2. Since \mathcal{F}|_{U_{\acute{e}tale}} and \mathcal{G}|_{U_{\acute{e}tale}} are quasi-coherent, so is their tensor product, see Descent, Lemma 35.10.3.
Proof of (5). Let \mathcal{F} and \mathcal{G} be in \textit{LQCoh}(\mathcal{O}_\mathcal {X}) with \mathcal{F} of finite presentation. Since (\mathit{Sch}/U)_{\acute{e}tale}= \mathcal{X}_{\acute{e}tale}/x is a localization of \mathcal{X}_{\acute{e}tale} at an object we see that the restriction of \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G}) to (\mathit{Sch}/U)_{\acute{e}tale} is equal to
\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}|_{(\mathit{Sch}/U)_{\acute{e}tale}}}( \mathcal{F}|_{(\mathit{Sch}/U)_{\acute{e}tale}}, \mathcal{G}|_{(\mathit{Sch}/U)_{\acute{e}tale}})
by Modules on Sites, Lemma 18.27.2. The morphism of ringed topoi (U_{\acute{e}tale}, \mathcal{O}_ U) \to ((\mathit{Sch}/U)_{\acute{e}tale}, \mathcal{O}) is flat as the pullback of \mathcal{O} is \mathcal{O}_ U. Hence the pullback of \mathcal{H} by this morphism is equal to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_{U_{\acute{e}tale}}, \mathcal{G}|_{U_{\acute{e}tale}}) by Modules on Sites, Lemma 18.31.4. In other words, the restriction of \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G}) to U_{\acute{e}tale} is \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_{U_{\acute{e}tale}}, \mathcal{G}|_{U_{\acute{e}tale}}). Since \mathcal{F}|_{U_{\acute{e}tale}} and \mathcal{G}|_{U_{\acute{e}tale}} are quasi-coherent, so is \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_{U_{\acute{e}tale}}, \mathcal{G}|_{U_{\acute{e}tale}}), see Descent, Lemma 35.10.3. We conclude as before.
\square
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