During this proof we work exclusively with sheaves on the small \'etale sites, and we use $i_*, i^{-1}, \ldots$ to denote pushforward and pullback of sheaves of abelian groups instead of $i_{small, *}, i_{small}^{-1}$. \medskip\noindent Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. By Lemma \ref{lemma-closed-immersion-rings} applied with $\mathcal{A} = \mathcal{O}_X$ and $\mathcal{G} = \mathcal{B} = \mathcal{O}_Z$ we see that $i_*i^*\mathcal{F} = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_Z$. By Lemma \ref{lemma-closed-immersion-ideals} we see that we have a short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0 $$ It follows from properties of the tensor product that $\mathcal{F} \otimes_{\mathcal{O}_X} i_*\mathcal{O}_Z = \mathcal{F}/\mathcal{I}\mathcal{F}$. This proves (1) (except that we omit the verification that the map is induced by the adjunction mapping). \medskip\noindent Let $\mathcal{G}$ be any $\mathcal{O}_Z$-module. By Lemma \ref{lemma-closed-immersion-push-pull} we see that $i^{-1}i_*\mathcal{G} = \mathcal{G}$. Hence to prove (2) we have to show that the canonical map $\mathcal{G} \otimes_{i^{-1}\mathcal{O}_X} \mathcal{O}_Z \to \mathcal{G}$ is an isomorphism. This follows from general properties of tensor products if we can show that $i^{-1}\mathcal{O}_X \to \mathcal{O}_Z$ is surjective. By Lemma \ref{lemma-closed-immersion-push-pull} it suffices to prove that $i_*i^{-1}\mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective. Since the surjective map $\mathcal{O}_X \to i_*\mathcal{O}_Z$ factors through this map we see that (2) holds. \medskip\noindent Finally we prove the most interesting part of the lemma, namely part (3). A closed immersion is quasi-compact and separated, see Lemmas \ref{lemma-closed-immersion-quasi-compact} and \ref{lemma-closed-immersion-separated}. Hence Lemma \ref{lemma-pushforward} applies and the pushforward of a quasi-coherent sheaf on $Z$ is indeed a quasi-coherent sheaf on $X$. Thus we obtain our functor $i^{QCoh}_* : \QCoh(\mathcal{O}_Z) \to \QCoh(\mathcal{O}_X)$. It is clear from part (2) that $i^{QCoh}_*$ is fully faithful since it has a left inverse, namely $i^*$. \medskip\noindent Now we turn to the description of the essential image of the functor $i_*$. It is clear that $\mathcal{I}(i_*\mathcal{G}) = 0$ for any $\mathcal{O}_Z$-module, since $\mathcal{I}$ is the kernel of the map $\mathcal{O}_X \to i_*\mathcal{O}_Z$ which is the map we use to put an $\mathcal{O}_X$-module structure on $i_*\mathcal{G}$. Next, suppose that $\mathcal{F}$ is any quasi-coherent $\mathcal{O}_X$-module such that $\mathcal{I}\mathcal{F} = 0$. Then we see that $\mathcal{F}$ is an $i_*\mathcal{O}_Z$-module because $i_*\mathcal{O}_Z = \mathcal{O}_X/\mathcal{I}$. Hence in particular its support is contained in $|Z|$. We apply Lemma \ref{lemma-closed-immersion-push-pull} to see that $\mathcal{F} \cong i_*\mathcal{G}$ for some $\mathcal{O}_Z$-module $\mathcal{G}$. The only small detail left over is to see why $\mathcal{G}$ is quasi-coherent. This is true because $\mathcal{G} \cong i^*\mathcal{F}$ by part (2) and Properties of Spaces, Lemma \ref{spaces-properties-lemma-pullback-quasi-coherent}.