  For a small simplicial category A, we prove that the homotopy colimit functor
from the category of simplicial diagrams on A to the category of simplicial
sets over the homotopy-coherent nerve of A provides a left Quillen equivalence
between the projective model structure on the former category and the covariant
model structure on the latter. We compare this Quillen equivalence to the
straightening-unstraightening equivalence previously established by Lurie,
where the left adjoint goes in the opposite direction. The existence of left
Quillen functors in both directions considerably simplifies the proof that
these constructions provide Quillen equivalences. The results of this paper
generalize those of part I, where A was an ordinary category. The proofs for a
simplicial category are more involved and can be read independently.
