The Stacks project

Proof. We may replace each of the algebraic stacks $\mathcal{X}_ j$ by a scheme $U_ j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 101.33.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 96.12.3. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a surjective smooth morphism. Let $x$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 96.19.10 there exists an étale covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x$, $x_ i$ live over schemes $V$, $V_ i$, that $\{ V_ i \to V\} $ is an étale covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. The restriction $x_ i^*\mathcal{F}|_{V_{i, {\acute{e}tale}}}$ is equal to the restriction of $f^*\mathcal{F}$ to $V_{i, {\acute{e}tale}}$, see Sheaves on Stacks, Lemma 96.9.3. Hence $x^*\mathcal{F}|_{V_{\acute{e}tale}}$ is a sheaf on the small étale site of $V$ which is quasi-coherent when restricted to $V_{i, {\acute{e}tale}}$ for each $i$. This implies that it is quasi-coherent (as desired), for example by Properties of Spaces, Lemma 66.29.6. $\square$