Localizations of the category of A_∞ categories and internal Homs over a ring

We show that, over an arbitrary commutative ring, the localizations of the categories of dg categories, of unital and of strictly unital A_∞ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. The same result is also proved at the ∞-categorical level in the strictly unital case. As an application, we provide a new proof of the existence of internal Homs for the homotopy category of dg categories in terms of the category of unital A_∞ functors, thus yielding a complete proof of a claim by Kontsevich and Keller.