The Stacks project

Lemma 59.102.2. With notation as above. Let $f : X \to Y$ be a morphism of $(\mathit{Sch}/S)_{ph}$. Then there are commutative diagrams of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{\epsilon _ X} & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_{ph}) \ar[d]^{\epsilon _ Y} \\ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{\acute{e}tale}) \ar[rr]^{f_{big, {\acute{e}tale}}} & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_{\acute{e}tale}) } \]

and

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{ph}) \ar[rr]_{f_{big, ph}} \ar[d]_{a_ X} & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_{ph}) \ar[d]^{a_ Y} \\ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \ar[rr]^{f_{small}} & & \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) } \]

with $a_ X = \pi _ X \circ \epsilon _ X$ and $a_ Y = \pi _ X \circ \epsilon _ X$.

Proof. The commutativity of the diagrams follows from the discussion in Topologies, Section 34.11. $\square$

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