The Stacks project

Proof. If $s$ is an element of the right hand side, then there exists an open covering $U = \bigcup _{j \in J} U_ j$ such that $s|_{U_ j}$ is a finite sum $\sum _{i \in I_ j} s_{ji}$ with $s_{ji} \in \mathcal{F}_ i(U_ j)$. Because $U$ is quasi-compact we may assume that the covering is finite, i.e., that $J$ is finite. Then $I' = \bigcup _{j \in J} I_ j$ is a finite subset of $I$. Clearly, $s$ is a section of the subsheaf $\bigoplus _{i \in I'} \mathcal{F}_ i$. The result follows from the fact that for a finite direct sum sheafification is not needed, see Lemma 17.3.2 above. $\square$