The question is related to the following MO question

(Co-)Limits and fibrations of DG-Categories?

My question is,

1. how to define the homotopy limit (and colimit) of a system of dg-categories (let's fix a universe and a base ring $k$, and work only with small things...), and

2. is there an explicit description of the homotopy category of the homotopy limit of dg categories $$ Ho(holim_{i\in I}\mathscr C_i)=? $$ Recall that the homotopy category $Ho(\mathscr C)$ of a dg category $\mathscr C$ is the category with the same objects as $\mathscr C$ and the hom group is the cohomology at degree 0 of the hom complex in $\mathscr C:$ $$ Hom_{Ho(\mathscr C)}(X,Y)=H^0(Hom_{\mathscr C}(X,Y)). $$ One can ask similar questions to "categories" enriched in simplicial sets, which is a slightly more general setting.

I understand (sort of) that there is a model category structure (due to Tabuada) on the category $dg-Cat$ of dg categories such that weak equivalences are what one expects (to be a bit precise, a functor $F:\mathscr C\to\mathscr D$ is a w.e. if $$ Hom_{\mathscr C}(X,Y)\to Hom_{\mathscr D}(FX,FY) $$ is a quasi-isomorphism of complexes, and $Ho(F):Ho(\mathscr C)\to Ho(\mathscr D)$ is essentially surjective). But I don't know how to use this model structure to define homotopy limits.

Maybe one uses cofibrant replacement and the naive $\otimes$-structure on $dg-Cat$ to define a $\otimes^{\mathbb L}$-structure (following Toen) and shows that it is closed, so that one has internal hom $R\mathscr Hom$ on $dg-Cat,$ with which one defines homotopy limits (and colimits) of dg-categories by universal properties. I'm not sure. Both references and direct explanations are appreciated.

The question is related to the following MO question

(Co-)Limits and fibrations of DG-Categories?

My question is,

1. how to define the homotopy limit (and colimit) of a system of dg-categories (let's fix a universe and a base ring $k$, and work only with small things...), and

2. is there an explicit description of the homotopy category of the homotopy limit of dg categories $$ Ho(holim_{i\in I}\mathscr C_i)=? $$ Recall that the homotopy category $Ho(\mathscr C)$ of a dg category $\mathscr C$ is the category with the same objects as $\mathscr C$ and the hom group is the cohomology at degree 0 of the hom complex in $\mathscr C:$ $$ Hom_{Ho(\mathscr C)}(X,Y)=H^0(Hom_{\mathscr C}(X,Y)). $$ One can ask similar questions to "categories" enriched in simplicial sets, which is a slightly more general setting.

I understand (sort of) that there is a model category structure (due to Tabuada) on the category $dg-Cat$ of dg categories such that weak equivalences are what one expects (to be a bit precise, a functor $F:\mathscr C\to\mathscr D$ is a w.e. if $$ Hom_{\mathscr C}(X,Y)\to Hom_{\mathscr D}(FX,FY) $$ is a quasi-isomorphism of complexes, and $Ho(F):Ho(\mathscr C)\to Ho(\mathscr D)$ is essentially surjective). But I don't know how to use this model structure to define homotopy limits.

Maybe one uses cofibrant replacement and the naive $\otimes$-structure on $dg-Cat$ to define a $\otimes^{\mathbb L}$-structure (following Toen) and shows that it is closed, so that one has internal hom $R\mathscr Hom$ on $dg-Cat,$ with which one defines homotopy limits (and colimits) of dg-categories by universal properties. I'm not sure. Both references and direct explanations are appreciated.