  Let $F$ be a non-archimedean local field. We prove that every Bernstein
component in the smooth dual of each inner form of the general linear group
$GL_n(F)$ is canonically in bijection with the extended quotient for the
action, given by Bernstein, of a finite group on a complex torus. For inner
forms of $SL_n(F)$ we prove that each Bernstein component is canonically in
bijection with the associated twisted extended quotient. In both cases, the
bijections satisfy naturality properties with respect to the tempered dual,
parabolic induction, central character, and the local Langlands correspondence.
