Lemma 38.34.13. Let $S$ be a scheme. Let $\mathit{Sch}_ h$ be a big h site containing $S$. Then $(\textit{Aff}/S)_ h$ is a site.

Proof. Reasoning as in the proof of Topologies, Lemma 34.4.9 it suffices to show that the collection of standard h coverings satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard h covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ a standard h covering $\{ T_{ij} \to T_ i\} _{j \in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a h covering (Lemma 38.34.8), $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. Thus $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard h covering. $\square$

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