The central sheaf of a Grothendieck category

The center $Z(\mathcal{A})$ of an abelian category $\mathcal{A}$ is the endomorphism ring of the identity functor on that category. A localizing subcategory of a Grothendieck category $\mathcal{C}$ is said to be stable if it is stable under essential extensions. The set ${\mathbf{L}}^{st}(\mathcal{C})$ of stable localizing subcategories of $\mathcal{C}$ is partially ordered under reverse inclusion. We show $\mathcal{L}\mapsto Z(\mathcal{C}\mathrm{/}\mathcal{L})$ defines a sheaf of commutative rings on ${\mathbf{L}}^{st}(\mathcal{C})$ with respect to finite coverings. When $\mathcal{C}$ is assumed to be locally noetherian, we also show that the sheaf condition holds for arbitrary coverings.