The Stacks project

Proof. Choose a scheme $U$ and a surjective smooth morphism $f : U \to \mathcal{X}$. Set $R = U \times _\mathcal {X} U$ so that we obtain a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces with the property that $\mathcal{X} = [U/R]$, see Algebraic Stacks, Lemma 94.16.2. We may and do replace $\mathcal{X}$ by $[U/R]$. By Sheaves on Stacks, Proposition 96.14.3 there is an equivalence

Let us construct a functor

by the following rule: if $\mathcal{F}$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then we set

where $\alpha$ is the isomorphism

where the outer two morphisms are the comparison maps. Note that $q_2(\mathcal{F})$ is quasi-coherent precisely because $\mathcal{F}$ is locally quasi-coherent and that we used (and needed) the flat base change property in the construction of the descent datum $\alpha$. We omit the verification that the cocycle condition (see Groupoids in Spaces, Definition 78.12.1) holds. Looking at the proof of Sheaves on Stacks, Proposition 96.14.3 we see that $q_2 \circ i$ is the quasi-inverse to $q_1$. We define $Q = q_1 \circ q_2$. Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and let $\mathcal{G}$ be an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. We have

where the first equality is Sheaves on Stacks, Lemma 96.14.4 and the second equality holds because $q_1 \circ i$ and $q_2$ are quasi-inverse equivalences of categories. The assertion $Q \circ i \cong \text{id}$ is a formal consequence of the fact that $i$ is fully faithful. $\square$