Lemma 6.33.3. Let X be a topological space. Let X = \bigcup U_ i be an open covering. Let (\mathcal{F}_ i, \varphi _{ij}) be a glueing data of sheaves of abelian groups, resp. sheaves of algebraic structures, resp. sheaves of \mathcal{O}-modules for some sheaf of rings \mathcal{O} on X. Then the construction in the proof of Lemma 6.33.2 above leads to a sheaf of abelian groups, resp. sheaf of algebraic structures, resp. sheaf of \mathcal{O}-modules.
Proof. This is true because in the construction the set of sections \mathcal{F}(W) over an open W is given as the equalizer of the maps
\xymatrix{ \prod _{i \in I} \mathcal{F}_ i(W \cap U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{i, j\in I} \mathcal{F}_ i(W \cap U_ i \cap U_ j) }
And in each of the cases envisioned this equalizer gives an object in the relevant category whose underlying set is the object considered in the cited lemma. \square
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