  For a small 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 nerve of A establishes a left Quillen equivalence between the projective
(or Reedy) model structure on the former category and the covariant model
structure on the latter. We compare this equivalence to a Quillen equivalence
in the opposite direction previously established by Lurie. From our results we
deduce that a categorical equivalence of simplicial sets induces a Quillen
equivalence on the corresponding over-categories, equipped with the covariant
model structures. Also, we show that versions of Quillen's Theorems A and B for
infinity-categories easily follow.
