**Proof.**
By Lemma 61.12.21 we may describe $i_{pro\text{-}\acute{e}tale}$ in terms of the morphism of sites $u : X_{app} \to Z_{app}$, $V \mapsto V \times _ X Z$. The first statement of the lemma follows from Sites, Lemmas 7.22.1 and 7.22.2 (but with the roles of $u$ and $v$ reversed).

Proof of (1). By the description of $i$ as the morphism of topoi associated to $v$ this holds by the construction, see Sites, Lemma 7.21.1.

Proof of (2). Since the functor $v$ sends coverings to coverings by Lemma 61.25.1 we see that the presheaf $\mathcal{G} : V \mapsto \mathcal{F}(v(V))$ is a separated presheaf (Sites, Definition 7.10.9). Hence the sheafification of $\mathcal{G}$ is $\mathcal{G}^+$, see Sites, Theorem 7.10.10. Next, let $V$ be a weakly contractible object of $Z_{app}$. Let $\mathcal{V} = \{ V_ i \to V\} _{i = 1, \ldots , n}$ be any covering in $Z_{app}$. Set $\mathcal{V}' = \{ \coprod V_ i \to V\} $. Since $v$ commutes with finite disjoint unions (as a left adjoint or by the construction) and since $\mathcal{F}$ sends finite disjoint unions into products, we see that

\[ H^0(\mathcal{V}, \mathcal{G}) = H^0(\mathcal{V}', \mathcal{G}) \]

(notation as in Sites, Section 7.10; compare with Étale Cohomology, Lemma 59.22.1). Thus we may assume the covering is given by a single morphism, like so $\{ V' \to V\} $. Since $V$ is weakly contractible, this covering can be refined by the trivial covering $\{ V \to V\} $. It therefore follows that the value of $\mathcal{G}^+ = i^{-1}\mathcal{F}$ on $V$ is simply $\mathcal{F}(v(V))$ and (2) is proved.

Proof of (3). Every object of $Z_{app}$ has a covering by weakly contractible objects (Lemma 61.13.4). By the above we see that we would have $i^{Sh}_!h_ V = h_{v(V)}$ for $V$ weakly contractible if $i^{Sh}_!$ existed. The existence of $i^{Sh}_!$ then follows from Sites, Lemma 7.24.1.

Proof of (4). Existence of $i_!$ follows in the same way by setting $i_!\mathbf{Z}_ V = \mathbf{Z}_{v(V)}$ for $V$ weakly contractible in $Z_{app}$, using similar for direct sums, and applying Homology, Lemma 12.29.6. Details omitted.

Proof of (5). Let $V$ be a contractible object of $Z_{app}$. Then $i^{-1}i^{Sh}_!h_ V = i^{-1}h_{v(V)} = h_{u(v(V))} = h_ V$. (It is a general fact that $i^{-1}h_ U = h_{u(U)}$.) Since the sheaves $h_ V$ for $V$ contractible generate $\mathop{\mathit{Sh}}\nolimits (Z_{app})$ (Sites, Lemma 7.12.5) we conclude $\text{id} \to i^{-1}i^{Sh}_!$ is an isomorphism. Similarly for the map $\text{id} \to i^{-1}i_!$. Then $(i^{-1}i_*\mathcal{H})(V) = i_*\mathcal{H}(v(V)) = \mathcal{H}(u(v(V))) = \mathcal{H}(V)$ and we find that $i^{-1}i_* \to \text{id}$ is an isomorphism.

The fully faithfulness statements of (6) now follow from Categories, Lemma 4.24.4.
$\square$

## Comments (0)