= Solution
An $A$-<module homomorphism> $h:M\to N$ induces, by <localization of a module>, compatible maps
$$
M_a\longrightarrow N_a,\qquad m/a^r\longmapsto h(m)/a^r
$$
on all <principal open subsets> $D(a)$. These glue to an $\mathcal O_X$-linear <morphism of sheaves> $\widetilde h:\widetilde M\to\widetilde N$.
Conversely, taking <global sections> of an $\mathcal O_X$-linear <morphism of sheaves> gives an $A$-<module homomorphism> $h:M\to N$, since $\Gamma(X,\widetilde M)=M$ and $\Gamma(X,\widetilde N)=N$. On $D(a)$, compatibility with restriction determines the image of $m/1$, and $\mathcal O_X$-linearity determines the image of $a^{-r}m$. It must be $h(m)/a^r$. Thus the <morphism of sheaves> is uniquely recovered from $h$. Both constructions preserve addition and composition, proving the <affine module-sheaf equivalence> on morphisms:
$$
\boxed{\operatorname{Hom}_{\mathcal O_X}(\widetilde M,\widetilde N)\simeq\operatorname{Hom}_A(M,N).}
$$
No finite-generation assumption is used.
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