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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