Suppose $B$ is an abelian category and $A$ is a full abelian subcategory of it, and the injection functor $i: A \rightarrow B$ is fully faithful with derived functor \begin{equation} Di:D^b A \rightarrow D^b B \end{equation} My question is about the essential image of $Di$. For a bounded complex $Y^*$ in $D^bB$, if each of the cohomology of this complex $H^k(Y^*)$ is isomorphic to an object in $A$, does there exist a bounded complex $X^*$ in $D^bA$ such that $i(X^*)$ is isomorphic to $Y^*$ in $D^bB$?
What about the case when $B$ is the abelian category of $\mathbb{Q}$-mixed Hodge structures, and $A$ is the abelian subcategory generated by Tate objects $\mathbb{Q}(n), n \in \mathbb{Z}$?