Lemma 73.4.11. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism in $(\textit{Spaces}/S)_{\acute{e}tale}$.

We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 73.4.7 and $i_ T$ as in Lemma 73.4.8.

The functor $X_{spaces, {\acute{e}tale}} \to T_{spaces, {\acute{e}tale}}$, $(U \to X) \mapsto (U \times _ X Y \to Y)$ is continuous and induces a morphism of sites

\[ f_{spaces, {\acute{e}tale}} : Y_{spaces, {\acute{e}tale}} \longrightarrow X_{spaces, {\acute{e}tale}} \]The corresponding morphism of small étale topoi is denoted

\[ f_{small} : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]We have $f_{small, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times _ X Y/Y)$.

We have a commutative diagram of morphisms of sites

\[ \xymatrix{ Y_{spaces, {\acute{e}tale}} \ar[d]_{f_{spaces, {\acute{e}tale}}} & (\textit{Spaces}/Y)_{\acute{e}tale}\ar[d]^{f_{big}} \ar[l]^-{\pi _ Y}\\ X_{spaces, {\acute{e}tale}} & (\textit{Spaces}/X)_{\acute{e}tale}\ar[l]_-{\pi _ X} } \]so that $f_{small} \circ \pi _ Y = \pi _ X \circ f_{big}$ as morphisms of topoi.

We have $f_{small} = \pi _ X \circ f_{big} \circ i_ Y = \pi _ X \circ i_ f$.

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