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 to take it in $D(A)$ or $D(C)$

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 $D(C)$