The Stacks project

Lemma 35.2.4. Let $S$ be a scheme. Let $S = \bigcup U_ i$ be an open covering. Any descent datum on quasi-coherent sheaves for the family $\mathcal{U} = \{ U_ i \to S\} $ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_ S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.

Proof. This follows immediately from Sheaves, Section 6.33 and the fact that being quasi-coherent is a local property, see Modules, Definition 17.10.1. $\square$

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