  This is the continuation of the study of differential graded (dg) vertex
algebras previously defined by the authors. The goal of this paper is to
construct a functor from the category of dg vertex Lie algebras to the category
of dg vertex algebras which is left adjoint to the forgetful functor. This
functor not only provides an abundant number of examples of dg vertex algebras,
but it is also an important step in constructing a homotopy theory in the
category of vertex algebras. Vertex Lie algebras were introduced as analogues
of vertex algebras, but in which we only consider the singular part of the
vertex operator map and the equalities it satisfies. In this paper, we extend
the definition of vertex Lie algebras to the dg setting. We construct a pair of
adjoint functors between the categories of dg vertex algebras and dg vertex Lie
algebras, which leads to the explicit construction of dg vertex (operator)
algebras. We will give examples based on the Virasoro algebra, the
Neveu-Schwarz algebra, and dg Lie algebras.
