In Bosch's textbook, Algebraic Geometry and Commutative Algebra, he claims, on pages 258-9 in Prop 4 that the functor $M\mapsto\tilde{M}$ from $A$-modules to $\mathcal{O}_X$-modules is exact ($\tilde{M}$ is given by $\tilde{M}(D(f))=M_f$ for distinguished sets $D(f)$).
To do so, in his proof, he remarks that given an exact sequence of $A$-modules $M'\to M \to M''$, "any section of the kernel of $\tilde{M}\to\tilde{M}''$ living on some open subset of $X$ admits local preimages in $\tilde{M}'$. But then it follows that the sequence $\tilde{M}\to \tilde{M}\to \tilde{M}''$ is exact." I don't really understand this part in the quotations -- how exactly does this imply the sequence of module sheaves is exact? In particular, do we not have to be careful about using the sheafification of the image presheaf (which is kinda ugly)?
I think there is another way of proving that the functor is exact (noting that we can look at the stalks instead), but I'd like to understand what Bosch means here.