Lemma 34.5.7. Let $\mathit{Sch}_{smooth}$ be a big smooth site as in Definition 34.5.6. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{smooth})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary smooth covering of $T$.

There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{smooth}$ which refines $\{ T_ i \to T\} _{i \in I}$.

If $\{ T_ i \to T\} _{i \in I}$ is a standard smooth covering, then it is tautologically equivalent to a covering of $\mathit{Sch}_{smooth}$.

If $\{ T_ i \to T\} _{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\mathit{Sch}_{smooth}$.

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