  We generalize the local-global compatibility result in arXiv:1506.04022 to
higher dimensional cases, by examining the relation between Scholze's functor
and cohomology of Kottwitz-Harris-Taylor type Shimura varieties. Along the way
we prove a cuspidality criterion from type theory. We also deal with
compatibility for torsion classes in the case of semisimple mod $p$ Galois
representations with distinct irreducible components under certain flatness
hypotheses.
