Localizations of the Category of A∞ Categories and Internal Homs

We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital A∞ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the A∞ category of A∞ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital A∞ functors between them.