In the paper "Finite dimensional algebras and highest weight categories" of Cline, Parshall and Scott is stated as follows:
Let $\mathcal{B}$ be an abelian subcategory of a triangulated category $\mathcal{D}.$ Let $\mathcal{D'}=\mathcal{D}_{\mathcal{B}}$ be the full triangulated subcategory of $\mathcal{D}$ generated by $\mathcal{B}$ (in the sense that $\mathcal{D'}$ is the smallest strict triangulated subcategory of $\mathcal{D}$ containing $\mathcal{B}$). Assume that the inclusion $\mathcal{B}\subset \mathcal{D}$ is induced by an exact functor $i_*:D^b(\mathcal{B})\rightarrow \mathcal{D}$ which factors through $\mathcal{D'}$ and which is a full embedding. Then $i_*$ induces an equivalence $D^b(\mathcal{B})\cong \mathcal{D'}=\mathcal{D}_{\mathcal{B}}$ of triangulated categories.
From definition of $\mathcal{D'}$ we know that $\mathcal{D'}\rightarrow \mathcal{D}$ is a full embedding and $D^b(\mathcal{B})\rightarrow \mathcal{D}$ is a full embedding. I think it's also neccesary to use the information that $i_*$ factors through $\mathcal{D'}.$ I can to conclude that $D^b(\mathcal{B})\rightarrow \mathcal{D}$ is embedding, but it is, of course, not sufficient to conclude natural equivalence. I don't know how to obtain equivalence of categories given above.
I would be grateful for your advices and hints.