The Stacks project

Lemma 34.4.19. Let $\mathit{Sch}_{\acute{e}tale}$ be a big ├ętale site. Consider a cartesian diagram

\[ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^ f \\ S' \ar[r]^ g & S } \]

in $\mathit{Sch}_{\acute{e}tale}$. Then $i_ g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.

Proof. Since the diagram is cartesian, we have for $U'/S'$ that $U' \times _{S'} T' = U' \times _ S T$. Hence both $i_ g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\mathit{Sch}/T)_{\acute{e}tale}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{S'} T')$ on $S'_{\acute{e}tale}$ (use Lemmas 34.4.13 and 34.4.16). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 7.28.1. $\square$

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