Suppose $X$ is a separated scheme, $F$ an abelian sheaf on $X$ in the Zariski topology.
In order to assign an $\mathcal{O}_X$-module structure on $F$, is it enough to do it on a single open affine cover of $X$, compatibly with pairwise intersections?
Or is it necessary to assign the $\mathcal{O}_X$-module structure on every affine open of $X$, compatibly with restriction maps?