Let $\mathcal A$ be an abelian category; $\mathcal B \subseteq \mathcal A$ a weak Serre subcategory. Does the inclusion $\mathbf D_{\mathcal B}(\mathcal A) \subseteq \mathbf D(\mathcal A)$ admit a right adjoint? If not in general, is there any condition on $\mathcal A$ and $\mathcal B$ that makes it true?