  We prove that the localizations of the categories of dg categories, of
cohomologically unital and strictly unital $A_\infty$ 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
$A_\infty$ category of $A_\infty$ 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
$A_\infty$ functors between them.
