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Lemma 73.4.13. Let $S$ be a scheme. Consider a cartesian diagram

\[ \xymatrix{ Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^ f \\ X' \ar[r]^ g & X } \]

in $(\textit{Spaces}/S)_{\acute{e}tale}$. 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 $U'/X'$ that $U' \times _{X'} Y' = U' \times _ X Y$. Hence both $i_ g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\textit{Spaces}/Y)_{\acute{e}tale}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{X'} Y')$ on $X'_{\acute{e}tale}$ (use Lemmas 73.4.7 and 73.4.10). 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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