On the derived DG functors

Assume that abelian categories $A,B$ over a field admit countable direct limits and that these limits are exact. Let $F:D_{dg}^{+}(A)-->D_{dg}^{+}(B)$ be a DG quasi-functor such that the functor $Ho(F):D^{+}(A)\to D^{+}(B)$ carries $D^{\ge 0}(A)$ to $D^{\ge 0}(B)$ and such that, for every $i>0$, the functor $H^{i}F:A\to B$ is effaceable. We prove that $F$ is canonically isomorphic to the right derived DG functor $R{H}^0(F)$. We also prove a similar result for bounded derived DG categories in a more general setting. We give an example showing that the corresponding statements for triangulated functors are false. We prove a formula that expresses Hochschild cohomology of the categories $D_{dg}^b(A)$, $D_{dg}^{+}(A)$ as the $Ext$ groups in the abelian category of left exact functors $A\to IndB$ .