The Stacks project

Proof. As $\textit{Mod}(\mathcal{O}_ X)$ is abelian (Lemma 17.3.1) it has all finite limits and colimits (Homology, Lemma 12.5.5). Thus the existence of limits and colimits and their description follows from the existence of products and coproducts and their description (see discussion above) and Categories, Lemmas 4.14.11 and 4.14.12. Since sheafification commutes with taking stalks we see that colimits commute with taking stalks. Part (3) signifies that given a system $0 \to \mathcal{F}_ i \to \mathcal{G}_ i \to \mathcal{H}_ i \to 0$ of exact sequences of $\mathcal{O}_ X$-modules over a directed set $I$ the sequence $0 \to \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i \to 0$ is exact as well. Since we can check exactness on stalks (Lemma 17.3.1) this follows from the case of modules which is Algebra, Lemma 10.8.8. We omit the proof of (4). $\square$