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