We have seen in Lemma \ref{lemma-quasi-coherent-gives-quasi-coherent} that the functor is well defined. It is straightforward to show that the functor is fully faithful (we omit the verification). To finish the proof we will show that a quasi-coherent $\mathcal{O}$-module on $(\Sch/S)_\tau$ gives rise to a descent datum for quasi-coherent sheaves relative to a $\tau$-covering of $S$. Having produced this descent datum we will appeal to Proposition \ref{proposition-fpqc-descent-quasi-coherent} to get the corresponding quasi-coherent sheaf on $S$. \medskip\noindent Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}$-modules on the big $\tau$ site of $S$. By Modules on Sites, Definition \ref{sites-modules-definition-site-local} there exists a $\tau$-covering $\{S_i \to S\}_{i \in I}$ of $S$ such that each of the restrictions $\mathcal{G}|_{(\Sch/S_i)_\tau}$ has a global presentation $$ \bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\Sch/S_i)_\tau} \longrightarrow 0 $$ for some index sets $J_i$ and $K_i$. We claim that this implies that $\mathcal{G}|_{(\Sch/S_i)_\tau}$ is $\mathcal{F}_i^a$ for some quasi-coherent sheaf $\mathcal{F}_i$ on $S_i$. Namely, this is clear for the direct sums $\bigoplus\nolimits_{k \in K_i} \mathcal{O}|_{(\Sch/S_i)_\tau}$ and $\bigoplus\nolimits_{j \in J_i} \mathcal{O}|_{(\Sch/S_i)_\tau}$. Hence we see that $\mathcal{G}|_{(\Sch/S_i)_\tau}$ is a cokernel of a map $\varphi : \mathcal{K}_i^a \to \mathcal{L}_i^a$ for some quasi-coherent sheaves $\mathcal{K}_i$, $\mathcal{L}_i$ on $S_i$. By the fully faithfulness of $(\ )^a$ we see that $\varphi = \phi^a$ for some map of quasi-coherent sheaves $\phi : \mathcal{K}_i \to \mathcal{L}_i$ on $S_i$. Then it is clear that $\mathcal{G}|_{(\Sch/S_i)_\tau} \cong \Coker(\phi)^a$ as claimed. \medskip\noindent Since $\mathcal{G}$ lives on all of the category $(\Sch/S)_\tau$ we see that $$ (\text{pr}_0^*\mathcal{F}_i)^a \cong \mathcal{G}|_{(\Sch/(S_i \times_S S_j))_\tau} \cong (\text{pr}_1^*\mathcal{F})^a $$ as $\mathcal{O}$-modules on $(\Sch/(S_i \times_S S_j))_\tau$. Hence, using fully faithfulness again we get canonical isomorphisms $$ \phi_{ij} : \text{pr}_0^*\mathcal{F}_i \longrightarrow \text{pr}_1^*\mathcal{F}_j $$ of quasi-coherent modules over $S_i \times_S S_j$. We omit the verification that these satisfy the cocycle condition. Since they do we see by effectivity of descent for quasi-coherent sheaves and the covering $\{S_i \to S\}$ (Proposition \ref{proposition-fpqc-descent-quasi-coherent}) that there exists a quasi-coherent sheaf $\mathcal{F}$ on $S$ with $\mathcal{F}|_{S_i} \cong \mathcal{F}_i$ compatible with the given descent data. In other words we are given $\mathcal{O}$-module isomorphisms $$ \phi_i : \mathcal{F}^a|_{(\Sch/S_i)_\tau} \longrightarrow \mathcal{G}|_{(\Sch/S_i)_\tau} $$ which agree over $S_i \times_S S_j$. Hence, since $\SheafHom_\mathcal{O}(\mathcal{F}^a, \mathcal{G})$ is a sheaf (Modules on Sites, Lemma \ref{sites-modules-lemma-internal-hom}), we conclude that there is a morphism of $\mathcal{O}$-modules $\mathcal{F}^a \to \mathcal{G}$ recovering the isomorphisms $\phi_i$ above. Hence this is an isomorphism and we win. \medskip\noindent The case of the sites $S_\etale$ and $S_{Zar}$ is proved in the exact same manner.