  We characterize the Local Langlands Correspondence (LLC) for inner forms of
$GL_n$ via the Jacquet-Langlands Correspondence (JLC) and compatibility with
the Langlands Classification. We show that LLC satisfies a natural
compatibility with parabolic induction and characterize LLC for inner forms as
a unique family of bijections $\Pi(GL_r(D)) \to \Phi(GL_r(D))$ for each $r$,
(for a fixed $D$) satisfying certain properties. We construct a surjective map
of Bernstein centers $\mathfrak{Z}(GL_n(F))\to \mathfrak{Z}(GL_r(D))$ and show
this produces pairs of matching distributions. Finally, we construct explicit
Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke
algebras of $GL_r(D)$, and thereby produce many explicit pairs of matching
functions.

Representation Theory