Proof.
The corresponding facts hold for quasi-coherent modules on the scheme $S$, see Schemes, Section 26.24. The proof will be to use Lemma 35.10.2 to transfer these truths to $S_{\acute{e}tale}$.
Proof of (1). Let $\mathcal{F}_ i$, $i \in I$ be a family of objects of $\mathit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$. Write $\mathcal{F}_ i = \mathcal{G}_ i^ a$ for some quasi-coherent modules $\mathcal{G}_ i$ on $S$. Then $\bigoplus \mathcal{F}_ i = (\bigoplus \mathcal{G}_ i)^ a$ by the lemma cited and we conclude.
Proof of (2). Let $\mathcal{I} \to \mathit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Write $\mathcal{F}_ i = \mathcal{G}_ i^ a$ so we get a diagram $\mathcal{I} \to \mathit{QCoh}(\mathcal{O}_ S)$. Then $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i = (\mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i)^ a$ by the lemma cited and we conclude.
Proof of (3). Let $a : \mathcal{F} \to \mathcal{F}'$ be an arrow of $\mathit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$. Write $a = b^ a$ for some map $b : \mathcal{G} \to \mathcal{G}'$ of quasi-coherent modules on $S$. By the lemma cited we have $\mathop{\mathrm{Ker}}(a) = \mathop{\mathrm{Ker}}(b)^ a$ and $\mathop{\mathrm{Coker}}(a) = \mathop{\mathrm{Coker}}(b)^ a$ and we conclude.
Proof of (4). This follows from (3) except in the case when we know $\mathcal{F}_1$ and $\mathcal{F}_3$ are quasi-coherent. In this case write $\mathcal{F}_1 = \mathcal{G}_1^ a$ and $\mathcal{F}_3 = \mathcal{G}_3^ a$ with $\mathcal{G}_ i$ quasi-coherent on $S$. By Lemma 35.10.2 part (10) we conclude.
Proof of (5). Let $\mathcal{F}$ and $\mathcal{F}'$ be in $\mathit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$. Write $\mathcal{F} = \mathcal{G}^ a$ and $\mathcal{F}' = (\mathcal{G}')^ a$ with $\mathcal{G}$ and $\mathcal{G}'$ quasi-coherent on $S$. By the lemma cited we have $\mathcal{F} \otimes _\mathcal {O} \mathcal{F}' = (\mathcal{G} \otimes _{\mathcal{O}_ S} \mathcal{G}')^ a$ and we conclude.
Proof of (6). Let $\mathcal{F}$ and $\mathcal{G}$ be in $\mathit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$ with $\mathcal{F}$ of finite presentation. Write $\mathcal{F} = \mathcal{H}^ a$ and $\mathcal{G} = (\mathcal{I})^ a$ with $\mathcal{H}$ and $\mathcal{I}$ quasi-coherent on $S$. By Lemma 35.8.10 we see that $\mathcal{H}$ is of finite presentation. By Lemma 35.10.2 part (8) we have $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) = (\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{H}, \mathcal{I}))^ a$ and we conclude.
$\square$
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