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Proof. This proof is a repeat of the proof in the case of schemes, see Properties, Proposition 28.23.4. We advise the reader to read that proof first.

Part (1) means $\mathit{QCoh}(\mathcal{O}_ X)$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section 19.10. By Lemma 66.29.7 colimits in $\mathit{QCoh}(\mathcal{O}_ X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_ X)$. By Modules on Sites, Lemma 18.14.2 filtered colimits are exact. Hence (a) and (b) hold.

To construct a generator, choose a presentation $X = U/R$ so that $(U, R, s, t, c)$ is an étale groupoid scheme and in particular $s$ and $t$ are flat morphisms of schemes. Pick a cardinal $\kappa$ as in Groupoids, Lemma 39.15.7. Pick a collection $(\mathcal{E}_ t, \alpha _ t)_{t \in T}$ of $\kappa$-generated quasi-coherent modules on $(U, R, s, t, c)$ as in Groupoids, Lemma 39.15.6. Let $\mathcal{F}_ t$ be the quasi-coherent module on $X$ which corresponds to the quasi-coherent module $(\mathcal{E}_ t, \alpha _ t)$ via the equivalence of categories of Proposition 66.32.1. Then we see that every quasi-coherent module $\mathcal{H}$ is the directed colimit of its quasi-coherent submodules which are isomorphic to one of the $\mathcal{F}_ t$. Thus $\bigoplus _ t \mathcal{F}_ t$ is a generator of $\mathit{QCoh}(\mathcal{O}_ X)$ and we conclude that (c) holds. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem 19.11.7 and Lemma 19.13.2.

Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_ X)$ we consider the functor

This functor transforms colimits into limits, hence is representable, see Injectives, Lemma 19.13.1. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_ X)$. By the Yoneda lemma (Categories, Lemma 4.3.5) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ is fully faithful. $\square$