The Stacks project

Proof. Proof of (1). Taking sections over $W$ commutes with filtered colimits with injective transition maps by Sites, Lemma 7.17.7. If $\mathcal{F}_ i$ is a family of sheaves of sets indexed by a set $I$. Then $\coprod \mathcal{F}_ i$ is the filtered colimit over the partially ordered set of finite subsets $E \subset I$ of the coproducts $\mathcal{F}_ E = \coprod _{i \in E} \mathcal{F}_ i$. Since the transition maps are injective we conclude.

Proof of (2). Let $\mathcal{F}_ i$ be a family of sheaves of $\mathcal{O}$-modules indexed by a set $I$. Then $\bigoplus \mathcal{F}_ i$ is the filtered colimit over the partially ordered set of finite subsets $E \subset I$ of the direct sums $\mathcal{F}_ E = \bigoplus _{i \in E} \mathcal{F}_ i$. A filtered colimit of abelian sheaves can be computed in the category of sheaves of sets. Moreover, for $E \subset E'$ the transition map $\mathcal{F}_ E \to \mathcal{F}_{E'}$ is injective (as sheafification is exact and the injectivity is clear on underlying presheaves). Hence it suffices to show the result for a finite index set by Sites, Lemma 7.17.7. The finite case is dealt with in Lemma 18.3.2 (it holds over any object of $\mathcal{C}$). $\square$