  In [19], Zheng studied the bounded derived categories of constructible
$\bar{\mathbb{Q}}_l$-sheaves on some algebraic stacks consisting of the
representations of a enlarged quiver and categorified the integrable highest
weight modules of the corresponding quantum group by using these categories. In
this paper, we shall generalize Zheng's work to highest weight modules of a
subalgebra of the double Ringel-Hall algebra associated to a quiver in a
functional version.
