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

where $U \in X_{\acute{e}tale}$, $V \in Y_{\acute{e}tale}$ such that whenever given an extended diagram

with $V' \to V$ and $U' \to U$ étale morphisms of schemes the diagram

commutes.

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