Does $\alpha$ determine a morphism of $\mathcal O_X$-modules $\mathcal F^{\dagger}\to \mathcal G^{\dagger}$?

Question

Let $\mathcal F,\mathcal G$ be two presheaves of modules on $X=\operatorname{Spec}A$, $\alpha:\mathcal F\to \mathcal G$ is a morphism compatible with restrictions, only defined on principal open sets of $X$, does $\alpha$ determine a morphism of $\mathcal O_X$-modules $\mathcal F^{\dagger}\to \mathcal G^{\dagger}$?