Lemma 66.21.3. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Then there is a canonical map f^\sharp : f_{small}^{-1}\mathcal{O}_ Y \to \mathcal{O}_ X such that
(f_{small}, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}), \mathcal{O}_ Y)
is a morphism of ringed topoi. Furthermore,
The construction f \mapsto (f_{small}, f^\sharp ) is compatible with compositions.
If f is a morphism of schemes, then f^\sharp is the map described in Descent, Remark 35.8.4.
Proof.
By Lemma 66.18.10 it suffices to give an f-map from \mathcal{O}_ Y to \mathcal{O}_ X. In other words, for every commutative diagram
\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} we have to give a map of rings (f^\sharp )_{(U, V, g)} : \Gamma (V, \mathcal{O}_ V) \to \Gamma (U, \mathcal{O}_ U). Of course we just take (f^\sharp )_{(U, V, g)} = g^\sharp . It is clear that this is compatible with restriction mappings and hence indeed gives an f-map. We omit checking compatibility with compositions and agreement with the construction in Descent, Remark 35.8.4.
\square
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