Let $\mathcal{A}$ be a subcategory of $\mathcal{C}$. Let $D(\mathcal{A})$ and $D(\mathcal{C})$ be the associated derived categories. We can define $D\_\mathcal{A}(\mathcal{C}) = \{X \in \mathcal{C}\hspace{2pt} |\hspace{2pt} H^i(X) \in \mathcal{A}\}$. We have natural inclusions $$D(\mathcal{A}) \to D\_\mathcal{A}(\mathcal{C}) \to D(\mathcal{C})$$


My question is this: When is $D(\mathcal{A}) \to D\_\mathcal{A}(\mathcal{C})$ an equivalence of categories? 


I have read that it is not always an equivalence, but that it often is. I have examples of when it is: Is there an easy example for when it isn't? 



Equivalence between a derived subcategory and a subcategory of the derived category
Take $C$ to be sheaves of abelian groups on the sphere, and let $A$ be the abelian subcategory of locally constant abelian groups. Then $A$ is equivalent to the category of abelian groups and so $Hom(\mathbb Z,\mathbb Z[2])$ is different depending on whether you take it in $D(A)$ or $D(C)$


