Definition 66.18.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$ and let $\mathcal{G}$ be a sheaf of sets on $Y_{\acute{e}tale}$. An $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$ is a collection of maps $\varphi _{(U, V, g)} : \mathcal{G}(V) \to \mathcal{F}(U)$ indexed by commutative diagrams
\[ \xymatrix{ U \ar[d]_ g \ar[r] & X \ar[d]^ f \\ V \ar[r] & Y } \]
where $U \in X_{\acute{e}tale}$, $V \in Y_{\acute{e}tale}$ such that whenever given an extended diagram
\[ \xymatrix{ U' \ar[r] \ar[d]_{g'} & U \ar[d]_ g \ar[r] & X \ar[d]^ f \\ V' \ar[r] & V \ar[r] & Y } \]
with $V' \to V$ and $U' \to U$ étale morphisms of schemes the diagram
\[ \xymatrix{ \mathcal{G}(V) \ar[rr]_{\varphi _{(U, V, g)}} \ar[d]_{\text{restriction of }\mathcal{G}} & & \mathcal{F}(U) \ar[d]^{\text{restriction of }\mathcal{F}} \\ \mathcal{G}(V') \ar[rr]^{\varphi _{(U', V', g')}} & & \mathcal{F}(U') } \]
commutes.
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