Let $S = \bigcup_{j \in J} V_j$ be an affine open covering. For $j, j' \in J$ we denote $V_{jj'} = V_j \cap V_{j'}$ the intersection (which need not be affine). For $V \subset S$ open we denote $\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}$ which is a fpqc-covering (Topologies, Lemma \ref{topologies-lemma-fpqc}). By definition of an fpqc covering, we can find for each $j \in J$ a finite set $K_j$, a map $\underline{i} : K_j \to I$, affine opens $U_{\underline{i}(k), k} \subset U_{\underline{i}(k)}$, $k \in K_j$ such that $\mathcal{V}_j = \{U_{\underline{i}(k), k} \to V_j\}_{k \in K_j}$ is a standard fpqc covering of $V_j$. And of course, $\mathcal{V}_j$ is a refinement of $\mathcal{U}_{V_j}$. Picture $$ \xymatrix{ \mathcal{V}_j \ar[r] \ar@{~>}[d] & \mathcal{U}_{V_j} \ar[r] \ar@{~>}[d] & \mathcal{U} \ar@{~>}[d] \\ V_j \ar@{=}[r] & V_j \ar[r] & S } $$ where the top horizontal arrows are morphisms of families of morphisms with fixed target (see Sites, Definition \ref{sites-definition-morphism-coverings}). \medskip\noindent To prove the proposition you show successively the faithfulness, fullness, and essential surjectivity of the functor from quasi-coherent sheaves to descent data. \medskip\noindent Faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent sheaves on $S$ and let $a, b : \mathcal{F} \to \mathcal{G}$ be homomorphisms of $\mathcal{O}_S$-modules. Suppose $\varphi_i^*(a) = \varphi^*(b)$ for all $i$. Pick $s \in S$. Then $s = \varphi_i(u)$ for some $i \in I$ and $u \in U_i$. Since $\mathcal{O}_{S, s} \to \mathcal{O}_{U_i, u}$ is flat, hence faithfully flat (Algebra, Lemma \ref{algebra-lemma-local-flat-ff}) we see that $a_s = b_s : \mathcal{F}_s \to \mathcal{G}_s$. Hence $a = b$. \medskip\noindent Fully faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent sheaves on $S$ and let $a_i : \varphi_i^*\mathcal{F} \to \varphi_i^*\mathcal{G}$ be homomorphisms of $\mathcal{O}_{U_i}$-modules such that $\text{pr}_0^*a_i = \text{pr}_1^*a_j$ on $U_i \times_U U_j$. We can pull back these morphisms to get morphisms $$ a_k : \mathcal{F}|_{U_{\underline{i}(k), k}} \longrightarrow \mathcal{G}|_{U_{\underline{i}(k), k}} $$ $k \in K_j$ with notation as above. Moreover, Lemma \ref{lemma-refine-descent-datum} assures us that these define a morphism between (canonical) descent data on $\mathcal{V}_j$. Hence, by Lemma \ref{lemma-standard-fpqc-covering}, we get correspondingly unique morphisms $a_j : \mathcal{F}|_{V_j} \to \mathcal{G}|_{V_j}$. To see that $a_j|_{V_{jj'}} = a_{j'}|_{V_{jj'}}$ we use that both $a_j$ and $a_{j'}$ agree with the pullback of the morphism $(a_i)_{i \in I}$ of (canonical) descent data to any covering refining both $\mathcal{V}_{j, V_{jj'}}$ and $\mathcal{V}_{j', V_{jj'}}$, and using the faithfulness already shown. For example the covering $\mathcal{V}_{jj'} = \{V_k \times_S V_{k'} \to V_{jj'}\}_{k \in K_j, k' \in K_{j'}}$ will do. \medskip\noindent Essential surjectivity. Let $\xi = (\mathcal{F}_i, \varphi_{ii'})$ be a descent datum for quasi-coherent sheaves relative to the covering $\mathcal{U}$. Pull back this descent datum to get descent data $\xi_j$ for quasi-coherent sheaves relative to the coverings $\mathcal{V}_j$ of $V_j$. By Lemma \ref{lemma-standard-fpqc-covering} once again there exist quasi-coherent sheaves $\mathcal{F}_j$ on $V_j$ whose associated canonical descent datum is isomorphic to $\xi_j$. By fully faithfulness (proved above) we see there are isomorphisms $$ \phi_{jj'} : \mathcal{F}_j|_{V_{jj'}} \longrightarrow \mathcal{F}_{j'}|_{V_{jj'}} $$ corresponding to the isomorphism of descent data between the pullback of $\xi_j$ and $\xi_{j'}$ to $\mathcal{V}_{jj'}$. To see that these maps $\phi_{jj'}$ satisfy the cocycle condition we use faithfulness (proved above) over the triple intersections $V_{jj'j''}$. Hence, by Lemma \ref{lemma-zariski-descent-effective} we see that the sheaves $\mathcal{F}_j$ glue to a quasi-coherent sheaf $\mathcal{F}$ as desired. We still have to verify that the canonical descent datum relative to $\mathcal{U}$ associated to $\mathcal{F}$ is isomorphic to the descent datum we started out with. This verification is omitted.