Lemma 34.4.16. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{\acute{e}tale}$.

We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 34.4.12 and $i_ T$ as in Lemma 34.4.13.

The functor $S_{\acute{e}tale}\to T_{\acute{e}tale}$, $(U \to S) \mapsto (U \times _ S T \to T)$ is continuous and induces a morphism of sites

\[ f_{small} : T_{\acute{e}tale}\longrightarrow S_{\acute{e}tale} \]We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$.

We have a commutative diagram of morphisms of sites

\[ \xymatrix{ T_{\acute{e}tale}\ar[d]_{f_{small}} & (\mathit{Sch}/T)_{\acute{e}tale}\ar[d]^{f_{big}} \ar[l]^{\pi _ T}\\ S_{\acute{e}tale}& (\mathit{Sch}/S)_{\acute{e}tale}\ar[l]_{\pi _ S} } \]so that $f_{small} \circ \pi _ T = \pi _ S \circ f_{big}$ as morphisms of topoi.

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

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