The Stacks project

Proof. Let $g : \mathcal{Y} \to \mathcal{X}$ be as in (1). Let $y$ be an object of $\mathcal{Y}$ lying over a scheme $V$. By Sheaves on Stacks, Lemma 96.9.3 we have $(g^*\mathcal{F})|_{V_{\acute{e}tale}} = \mathcal{F}|_{V_{\acute{e}tale}}$. Moreover a comparison mapping for the sheaf $g^*\mathcal{F}$ on $\mathcal{Y}$ is a special case of a comparison map for the sheaf $\mathcal{F}$ on $\mathcal{X}$, see Sheaves on Stacks, Lemma 96.9.3. In this way (1) is clear.

Proof of (2). We use the characterization of weak Serre subcategories of Homology, Lemma 12.10.3. Kernels and cokernels of maps between sheaves having the flat base change property also have the flat base change property. This is clear because $f_{small}^*$ is exact for a flat morphism of schemes and since the restriction functors $(-)|_{U_{\acute{e}tale}}$ are exact (because we are working in the étale topology). Finally, if $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is a short exact sequence of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and the outer two sheaves have the flat base change property then the middle one does as well, again because of the exactness of $f_{small}^*$ and the restriction functors (and the 5 lemma).

Proof of (3). Let $f_ i : \mathcal{X}_ i \to \mathcal{X}$ be a jointly surjective family of smooth morphisms of algebraic stacks and assume each $f_ i^*\mathcal{F}$ has the flat base change property. By part (1), the definition of an algebraic stack, and the fact that compositions of smooth morphisms are smooth (see Morphisms of Stacks, Lemma 101.33.2) we may assume that each $\mathcal{X}_ i$ is representable by a scheme. Let $\varphi : x \to x'$ be a morphism of $\mathcal{X}$ lying over a flat morphism $a : U \to U'$ of schemes. By Sheaves on Stacks, Lemma 96.19.10 there exists a jointly surjective family of étale morphisms $U'_ i \to U'$ such that $U'_ i \to U' \to \mathcal{X}$ factors through $\mathcal{X}_ i$. Thus we obtain commutative diagrams

Note that each $a_ i$ is a flat morphism of schemes as a base change of $a$. Denote $\psi _ i : x_ i \to x'_ i$ the morphism of $\mathcal{X}_ i$ lying over $a_ i$ with target $x_ i'$. By assumption the comparison maps $c_{\psi _ i} : (a_ i)_{small}^*\big (f_ i^*\mathcal{F}|_{(U'_ i)_{\acute{e}tale}}\big ) \to f_ i^*\mathcal{F}|_{(U_ i)_{\acute{e}tale}}$ is an isomorphism. Because the vertical arrows $U_ i' \to U'$ and $U_ i \to U$ are étale, the sheaves $f_ i^*\mathcal{F}|_{(U_ i')_{\acute{e}tale}}$ and $f_ i^*\mathcal{F}|_{(U_ i)_{\acute{e}tale}}$ are the restrictions of $\mathcal{F}|_{U'_{\acute{e}tale}}$ and $\mathcal{F}|_{U_{\acute{e}tale}}$ and the map $c_{\psi _ i}$ is the restriction of $c_\varphi $ to $(U_ i)_{\acute{e}tale}$, see Sheaves on Stacks, Lemma 96.9.3. Since $\{ U_ i \to U\} $ is an étale covering, this implies that the comparison map $c_\varphi $ is an isomorphism which is what we wanted to prove.

Proof of (4). Let $\mathcal{I} \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$, $i \mapsto \mathcal{F}_ i$ be a diagram and assume each $\mathcal{F}_ i$ has the flat base change property. Let $\varphi : x \to x'$ be a morphism of $\mathcal{X}$ lying over the flat morphism of schemes $f : U \to U'$. Recall that $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ is the sheafification of the presheaf colimit. As we are using the étale topology, it is clear that

and similarly for the restriction to $U'_{\acute{e}tale}$. Hence

For the second equality we used that $f_{small}^*$ commutes with colimits (as a left adjoint). The arrow is an isomorphism as each $\mathcal{F}_ i$ has the flat base change property. Thus the colimit has the flat base change property and (4) is true.

Part (5) holds because tensor products commute with pullbacks, see Modules on Sites, Lemma 18.26.2. Details omitted.

Let $\mathcal{F}$ and $\mathcal{G}$ be as in (6). Since $\mathcal{F}$ is quasi-coherent it has the flat base change property by Sheaves on Stacks, Lemma 96.12.2. Let $\varphi : x \to x'$ be a morphism of $\mathcal{X}$ lying over the flat morphism of schemes $f : U \to U'$. As we are using the étale topology, we have

and similarly for the restriction to $U'_{\acute{e}tale}$ (details omitted). Hence

Here the second equality is Modules on Sites, Lemma 18.31.4 which uses that $f : U \to U'$ is flat and hence the morphism of ringed sites $f_{small}$ is flat too. The arrow is an isomorphism as both $\mathcal{F}$ and $\mathcal{G}$ have the flat base change property. Thus our $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ has the flat base change property too as desired. $\square$