Lemma 84.8.4. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Then there are commutative diagrams of topoi
\xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{\epsilon _ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{ph}) \ar[d]^{\epsilon _ Y} \\ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \ar[rr]^{f_{big, {\acute{e}tale}}} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{\acute{e}tale}) }
and
\xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{a_ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{ph}) \ar[d]^{a_ Y} \\ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \ar[rr]^{f_{small}} & & \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) }
with a_ X = \pi _ X \circ \epsilon _ X and a_ Y = \pi _ X \circ \epsilon _ X.
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