The Stacks project

Proof. Part (2) holds for augmentations of simplicial objects in any category. Thus $Y(\varphi )^{-1} a_ m^{-1} \mathcal{F} = a_ n^{-1}\mathcal{F}$ which defines an $Y(\varphi )$-map from $a_ m^{-1}\mathcal{F}$ to $a_ n^{-1}\mathcal{F}$. Thus we can use (1) as the definition of $a^{-1}\mathcal{F}$ (using Lemma 85.2.2) and (4) as the definition of $a_*$. If this defines a morphism of topoi then part (3) follows because we'll have $g_ n^{-1} \circ a^{-1} = a_ n^{-1}$ by construction. To check $a$ is a morphism of topoi we have to show that $a^{-1}$ is left adjoint to $a_*$ and we have to show that $a^{-1}$ is exact. The last fact is immediate from the exactness of the functors $a_ n^{-1}$.

Let $\mathcal{F}$ be an object of $\mathop{\mathit{Sh}}\nolimits (X)$ and let $\mathcal{G}$ be an object of $\mathop{\mathit{Sh}}\nolimits (Y_{Zar})$. Given $\beta : a^{-1}\mathcal{F} \to \mathcal{G}$ we can look at the components $\beta _ n : a_ n^{-1}\mathcal{F} \to \mathcal{G}_ n$. These maps are adjoint to maps $\beta _ n : \mathcal{F} \to a_{n, *}\mathcal{G}_ n$. Compatibility with the simplicial structure shows that $\beta _0$ maps into $a_*\mathcal{G}$. Conversely, suppose given a map $\alpha : \mathcal{F} \to a_*\mathcal{G}$. For any $n$ choose a $\varphi : [0] \to [n]$. Then we can look at the composition

These are adjoint to maps $a_ n^{-1}\mathcal{F} \to \mathcal{G}_ n$ which define a morphism of sheaves $a^{-1}\mathcal{F} \to \mathcal{G}$. We omit the proof that the constructions given above define mutually inverse bijections

This finishes the proof. An interesting observation is here that this morphism of topoi does not correspond to any obvious geometric functor between the sites defining the topoi. $\square$