The Stacks project

Proof. This lemma follows from a purely sheaf theoretic statement discussed in Sites, Remark 7.26.7. We also give a direct proof in this case.

Given a sheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ we set $\mathcal{F}_ T = i_ p^{-1}\mathcal{F}$ where $p : T \to S$ is the structure morphism. Note that $\mathcal{F}_ T(U) = \mathcal{F}(U'/S)$ for any open $U \subset T$, and $U' \to T$ an open immersion in $(\mathit{Sch}/T)_{Zar}$ with image $U$, see Lemmas 34.3.12 and 34.3.13. Hence given $f : T' \to T$ over $S$ and $U, U' \to T$ we get a canonical map $\mathcal{F}_ T(U) = \mathcal{F}(U'/S) \to \mathcal{F}(U'\times _ T T'/S) = \mathcal{F}_{T'}(f^{-1}(U))$ where the middle is the restriction map of $\mathcal{F}$ with respect to the morphism $U' \times _ T T' \to U'$ over $S$. The collection of these maps are compatible with restrictions, and hence define an $f$-map $c_ f$ from $\mathcal{F}_ T$ to $\mathcal{F}_{T'}$, see Sheaves, Definition 6.21.7 and the discussion surrounding it. It is clear that $c_{f \circ g}$ is the composition of $c_ f$ and $c_ g$, since composition of restriction maps of $\mathcal{F}$ gives restriction maps.

Conversely, given a system $(\mathcal{F}_ T, c_ f)$ as in the lemma we may define a presheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ by simply setting $\mathcal{F}(T/S) = \mathcal{F}_ T(T)$. As restriction mapping, given $f : T' \to T$ we set for $s \in \mathcal{F}(T)$ the pullback $f^*(s)$ equal to $c_ f(s)$ (where we think of $c_ f$ as an $f$-map again). The condition on the $c_ f$ guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse. $\square$