Toen in [The homotopy theory of dg-categories and derived Morita theory](http://arxiv.org/abs/math/0408337) Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
RHom(C,D)=Int((C\otimes^{\mathbb{L}}D^{op})-Mod^{rqr}).
$$
where "rqr" stands for right quasi-representable and "Int" stands for the full subcategory consisting of cofibrant and fibrant objects.


On the other hand we can consider the dg-category of dg-functors between $C$ and $D$ and denote it by $DG(C,D)$. See Keller's [On differential graded categories](https://atlas.mat.ub.edu/grgta/articles/Keller.pdf) Section 2.3.


In general $DG(C,D)$ will be different from $RHom(C,D)$ (they are not even quasi-equivalent.) Nevertheless, is it true that their homotopy categories are equivalent? In other words, do we have
$$
H^0(DG(C,D))\cong H^0(RHom(C,D))?
$$
If not, do we have counter-examples?



Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?
Toen proved that $RHom$ provides the internal hom in the homotopy category of dg-categories. For what you want to be true, you need something more than this: you need to know that $RHom$ is actually the right-derived functor of the strict internal hom $DG$. Apparently, this was proved in [arXiv:1202.3359](http://arxiv.org/abs/1202.3359) (Prop. 5.2).


On the other hand, if you are willing to work up to derived Morita equivalence instead of quasi-equivalence, then there is the paper [G. Tabuada, *Homotopy theory of dg categories via localizing pairs and Drinfeld’s DG quotient*, HHA, 12 (2010), no. 1, p. 187-219]. Tabuada constructs a model category of localization pairs, which is Quillen equivalent to the Morita model structure on dg-categories, and shows that it admits a right-derived internal hom functor which agrees with Toen's construction.


