The Stacks project

Lemma 85.2.7. Let $f : Y \to X$ be a morphism of simplicial spaces. Then

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (Y_ n) \ar[d] \ar[r]_{f_ n} & \mathop{\mathit{Sh}}\nolimits (X_ n) \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (Y_{Zar}) \ar[r]^{f_{Zar}} & \mathop{\mathit{Sh}}\nolimits (X_{Zar}) } \]

is a commutative diagram of topoi.

Proof. Direct from the description of pullback functors in Lemmas 85.2.4 and 85.2.5. $\square$

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