Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-morphisms dg-functors, and $2$-morphisms (degree $0$, closed) natural transformations between dg-functors.
My question is: does the Dwyer-Kan model structure make dgCat a model $2$-category in the same of this post?
No. If dgCat were a model 2-category, then the 2-functor from dgCat to Cat that sends a dg-category $A$ to its underlying category (which has the same objects as $A$, and whose morphisms are the $0$-cycles in the hom chain complexes of $A$) would send DK-equivalences to equivalences of categories, since it is represented by the cofibrant "unit" dg-category $\mathbf{1}$ (which has a single object, and whose single hom chain complex is the group of integers $\mathbb{Z}$ concentrated in degree $0$) and since every dg-category is fibrant. But this is false: for example, consider the "two-object suspension" (i.e. the functor that sends a chain complex $C$ to the dg-category with two objects $\bot,\top$, and whose hom-object from $\bot$ to $\top$ is $C$, all other hom-objects being trivial, either $0$ or $\mathbf{1}$) of any quasi-isomorphism of chain complexes that is not bijective on $0$-cycles, such as the canonical morphism $\mathrm{Cyl}(\mathbf{1}) \to \mathbf{1}$.
