The Stacks project

Proof. Proof of (1). 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 quasi-coherent, see Modules on Sites, Lemma 18.23.4. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a smooth surjective morphism. Let $x : V \to \mathcal{X}$ 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 $x_ i$ lives over a scheme $V_ i$, that $\{ V_ i \to V\}$ is an étale covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. Then $x_ i^*\mathcal{F} = u_ i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 96.11.4 we see that $x^*\mathcal{F}$ is an fppf sheaf and since $x$ was arbitrary we see that $\mathcal{F}$ is a sheaf in the fppf topology. Applying Sheaves on Stacks, Lemma 96.11.3 we see that $\mathcal{F}$ is quasi-coherent.

Proof of (2). This is proved using exactly the same argument, which we fully write out here. 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 flat and locally finite presented morphisms are preserved by composition, see Morphisms of Stacks, Lemmas 101.25.2 and 101.27.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 96.11.2. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a surjective, flat, and locally finitely presented morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 96.19.10 there exists an fppf 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 $x_ i$ lives over a scheme $V_ i$, that $\{ V_ i \to V\}$ is an fppf covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. Then $x_ i^*\mathcal{F} = u_ i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 96.11.3 we see that $\mathcal{F}$ is quasi-coherent. $\square$