Let $0 \rightarrow E' \rightarrow E \rightarrow E''$ be a short exact sequence of quasi coherent sheaves on a scheme X. Show that the sequence $0 \rightarrow E'(X) \rightarrow E(X) \rightarrow E''(X)$ of global sections is left exact.
Considering the sequence as first a short exact sequence of abelian groups, I have managed to show that the sequence of sections must be left exact as a sequence of abelian groups. I thought of using this and then showing that the maps are then naturally maps of $O_X$ modules. (I might be completely misunderstanding how this is supposed to work. It's the best guess I have).
I know that there is a natural functor from the category of $O_X$ -modules to the category of abelian groups. And since the sequence $0 \rightarrow E' \rightarrow E \rightarrow E''$ is a sequence of quasi coherent sheaves, we have that given an affine open $U = Spec A \subseteq X$, the sequence $0 \rightarrow E'|_U \rightarrow E|_U \rightarrow E''|_U \rightarrow 0$ is of the form $0 \rightarrow \tilde {M'} \rightarrow \tilde {M} \rightarrow \tilde {M''} \rightarrow 0$ for some short exact sequence of A-modules $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$. But I'm not sure how to proceed.
I understand that the case for abelian groups is stronger in that it applies to a more general case. But from what I understand, it's also weaker as there is additional structure (Here, of an $O_X$ module) that it does not account for.
Any help would be appreciated.