The Stacks project

Lemma 85.2.5. Let $X$ be a simplicial space. The functor $X_{n, Zar} \to X_{Zar}$, $U \mapsto U$ is continuous and cocontinuous. The associated morphism of topoi $g_ n : \mathop{\mathit{Sh}}\nolimits (X_ n) \to \mathop{\mathit{Sh}}\nolimits (X_{Zar})$ satisfies

1. $g_ n^{-1}$ associates to the sheaf $\mathcal{F}$ on $X$ the sheaf $\mathcal{F}_ n$ on $X_ n$,

2. $g_ n^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X_ n)$ has a left adjoint $g^{Sh}_{n!}$,

3. $g^{Sh}_{n!}$ commutes with finite connected limits,

4. $g_ n^{-1} : \textit{Ab}(X_{Zar}) \to \textit{Ab}(X_ n)$ has a left adjoint $g_{n!}$, and

5. $g_{n!}$ is exact.