The Stacks project

Remark 111.32.8. Let $X$ be a topological space. In the category of abelian sheaves the direct sum of a family of sheaves $\{ {\mathcal F}_ i\} _{i\in I}$ is the sheaf associated to the presheaf $U \mapsto \oplus {\mathcal F}_ i(U)$. Consequently the stalk of the direct sum at a point $x$ is the direct sum of the stalks of the ${\mathcal F}_ i$ at $x$.