For a small category $$\mathbf{A}$$ A , we prove that the homotopy colimit functor from the category of simplicial diagrams on $$\mathbf{A}$$ A to the category of simplicial sets over the nerve of $$\mathbf{A}$$ 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 a version of Quillen’s Theorem A for $$\infty $$ ∞ -categories easily follows.