Lemma 59.103.2. With notation as above. Let $f : X \to Y$ be a morphism of $(\mathit{Sch}/S)_ h$. Then there are commutative diagrams of topoi

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

and

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

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

Proof. The commutativity of the diagrams follows similarly to what was said in Topologies, Section 34.11. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).