The Stacks project

Proof. Recall that $i_{small}^{-1}$ and $i_{small, *}$ is an adjoint pair of exact functors such that $i_{small}^{-1}i_{small, *}$ is isomorphic to the identify functor on abelian sheaves. See Proposition 59.46.4 and Lemma 59.36.2. Thus $i_{small, *} : D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is fully faithful and $i_{small}^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_ Z(X_{\acute{e}tale})$ and consider the adjunction map $K \to i_{small, *}i_{small}^{-1}K$. Using exactness of $i_{small, *}$ and $i_{small}^{-1}$ this induces the adjunction maps $H^ n(K) \to i_{small, *}i_{small}^{-1}H^ n(K)$ on cohomology sheaves. Since these cohomology sheaves are supported on $Z$ we see these adjunction maps are isomorphisms and we conclude that $D(Z_{\acute{e}tale}) \to D_ Z(X_{\acute{e}tale})$ is an equivalence.

To finish the proof we have to show that $R\mathcal{H}_ Z(K) = i_{small}^{-1}K$ if $K$ is an object of $D_ Z(X_{\acute{e}tale})$. To do this we can use that $K = i_{small, *}i_{small}^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet $ for $i_{small}^{-1}K$. Since $i_{small, *}$ is the right adjoint to the exact functor $i_{small}^{-1}$, the complex $i_{small, *}\mathcal{I}^\bullet $ is K-injective (Derived Categories, Lemma 13.31.9). We see that $R\mathcal{H}_ Z(K)$ is computed by $\mathcal{H}_ Z(i_{small, *}\mathcal{I}^\bullet ) = \mathcal{I}^\bullet $ as desired. $\square$