Checking quasi-coherence on an affine cover

The easy direction: assume $X={\bigcup }_i{U}_i$ is an affine open cover with $\mathcal{F}{\mathrm{\mid }}_{U_i}\cong \widetilde{M_i}$ for ${\mathcal{O}}_X(U_i)$-modules $M_i$. Every point of $X$ lies in some $U_i$, and $U_i$ is an affine open neighbourhood on which $\mathcal{F}$ is associated to a module, so the definition [F1] is satisfied and $\mathcal{F}$ is quasi-coherent. If $X=\mathrm{\varnothing }$ the empty cover applies and the only ${\mathcal{O}}_X$-module is the zero sheaf, which is quasi-coherent.

The converse, chart by chart: assume $\mathcal{F}$ quasi-coherent and let $U=\mathrm{Spec}A$ be an affine open of $X$. By [F1] the restriction $\mathcal{F}{\mathrm{\mid }}_U$ is quasi-coherent on the affine scheme $U$, so by [F2] the canonical comparison morphism ${\kappa }_U:\widetilde{\mathrm{\Gamma }(U,\mathcal{F})}\to \mathcal{F}{\mathrm{\mid }}_U$ is an isomorphism; in particular, for the members of any affine open cover of $X$ this exhibits $\mathcal{F}{\mathrm{\mid }}_{U_i}\cong \widetilde{\mathrm{\Gamma }(U_i,\mathcal{F})}$, proving the remaining direction of the equivalence.

Compatibility with restriction: let $W=\mathrm{Spec}C\subseteq U=\mathrm{Spec}A$ be affine open, with ring map $A\to C$, and let ${\kappa }_W:\widetilde{\mathrm{\Gamma }(W,\mathcal{F})}\to \mathcal{F}{\mathrm{\mid }}_W$ be the isomorphism of step 1.2. By [F3] applied to the $A$-module $\mathrm{\Gamma }(U,\mathcal{F})$ there is a canonical isomorphism $(\widetilde{\mathrm{\Gamma }(U,\mathcal{F})}){\mathrm{\mid }}_W\cong \widetilde{(C{\otimes }_A\mathrm{\Gamma }(U,\mathcal{F}))}$, and by naturality of $\kappa$ in [F2] the composite $(\widetilde{\mathrm{\Gamma }(U,\mathcal{F})}){\mathrm{\mid }}_W\to \mathcal{F}{\mathrm{\mid }}_W$ is obtained from the restriction map $\mathrm{\Gamma }(U,\mathcal{F})\to \mathrm{\Gamma }(W,\mathcal{F})$; under the identification $C{\otimes }_A\mathrm{\Gamma }(U,\mathcal{F})\cong \mathrm{\Gamma }(W,\mathcal{F})$ of [F3] (obtained by taking sections over $W$ of the restriction isomorphism followed by ${\kappa }_U{\mathrm{\mid }}_W$; it sends $c\otimes m$ to $c\cdot m{\mathrm{\mid }}_W$) this composite is precisely ${\kappa }_W$, so the displayed square commutes.

Overlaps and choice accounting: if $U,U^{\mathrm{\prime }}$ are affine opens and $W\subseteq U\cap U^{\mathrm{\prime }}$ is affine, step 2.1 applied to the inclusions $W\subseteq U$ and $W\subseteq U^{\mathrm{\prime }}$ shows that the identifications $\mathcal{F}{\mathrm{\mid }}_U\cong \widetilde{\mathrm{\Gamma }(U,\mathcal{F})}$ and $\mathcal{F}{\mathrm{\mid }}_{U^{\mathrm{\prime }}}\cong \widetilde{\mathrm{\Gamma }(U^{\mathrm{\prime }},\mathcal{F})}$ restrict to the same identification on $W$, since both are the canonical comparison ${\kappa }_W$; hence the identifications are compatible on overlaps. All data used are the restriction maps of $\mathcal{F}$ and the canonical comparisons, so no choice is made beyond the Axiom of Choice inherited through [F2] and [F3] and recorded in the Statement.