Lemma 59.48.5. Let $f : X \to Y$ be a closed immersion of schemes. Let $\tau \in \{ syntomic, smooth, {\acute{e}tale}\} $. The functor $V \mapsto X \times _ Y V$ defines an almost cocontinuous functor (see Sites, Definition 7.42.3) $(\mathit{Sch}/Y)_\tau \to (\mathit{Sch}/X)_\tau $ between big $\tau $ sites.
Proof. We have to show the following: given a morphism $V \to Y$ and any syntomic (resp. smooth, resp. étale) covering $\{ U_ i \to X \times _ Y V\} $, there exists a smooth (resp. smooth, resp. étale) covering $\{ V_ j \to V\} $ such that for each $j$, either $X \times _ Y V_ j$ is empty, or $X \times _ Y V_ j \to Z \times _ Y V$ factors through one of the $U_ i$. This follows on applying Lemma 59.48.4 above to the closed immersion $X \times _ Y V \to V$. $\square$
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