The Stacks project

Lemma 34.12.2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$. Suppose given big sites $\mathit{Sch}_\tau$ and $\mathit{Sch}'_\tau$. Assume that $\mathit{Sch}_\tau$ is contained in $\mathit{Sch}'_\tau$. The inclusion functor $\mathit{Sch}_\tau \to \mathit{Sch}'_\tau$ satisfies the assumptions of Sites, Lemma 7.21.8. There are morphisms of topoi

such that $f \circ g \cong \text{id}$. For any object $S$ of $\mathit{Sch}_\tau$ the inclusion functor $(\mathit{Sch}/S)_\tau \to (\mathit{Sch}'/S)_\tau$ satisfies the assumptions of Sites, Lemma 7.21.8 also. Hence similarly we obtain morphisms

with $f \circ g \cong \text{id}$.