Let $A$ be a commutative ring with identity, $X = \text{Spec}A$. Let $\mathscr F$ be a sheaf of $\mathscr O_X$ modules. Let $M = \Gamma(X, \mathscr F)$.
Is there a natural way to induce a morphism of $\mathscr O_X$-module $\widetilde M \to \mathscr F$, where $\widetilde M$ is the $\mathscr O_X$-modules given by $M$?
As $\mathscr F $ is a presheaf you have restriction morphisms $M \rightarrow \Gamma(U,\mathscr F)$ for every open subset $U$ of $X$ and if you take $U = D(f)$ a basic open subset, then this restriction map factors through $M_{(f)}$ and gives a map $M_{(f)} \rightarrow \Gamma(D(f),\mathscr F)$. These maps are compatible, and give rise to a morphism $\widetilde M \to \mathscr F$, has the open sets are a basis of $X$'s topology. Note that $\widetilde M$ will be a quasi-coherent $\mathscr O_X$-module, so that this morphism will not necessarily be surjective.
Note that there are projective $A$-modules $M$ which are not of finite type such that $\widetilde M$ is not a locally free $\mathscr O_X$-module. See Lazard's example given, as far as I remember, in the new edition of EGA 1.