Algebroid prestacks and deformations of ringed spaces

For a ringed space $(X,\mathcal{O})$, we show that the deformations of the abelian category $\mathsf{Mod}(\mathcal{O})$ of sheaves of $\mathcal{O}$-modules (Lowen and Van den Bergh, 2006) are obtained from algebroid prestacks, as introduced by Kontsevich. In case $X$ is a quasi-compact separated scheme the same is true for $\ensuremath{\mathsf{Qch}}(\mathcal{O})$, the category of quasi-coherent sheaves on $X$. It follows in particular that there is a deformation equivalence between $\mathsf{Mod}(\mathcal{O})$ and $\mathsf{Qch}(\mathcal{O})$.