  In \cite{rupel3},the authors defined algebra homomorphisms from the dual
Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an
appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the
representation category of an acyclic quiver, we give an alternative proof by
using the cluster multiplication formulas in \cite{DX}. Moreover, if the
underlying graph of $Q$ is bipartite and the matrix $B$ associated to the
quiver $Q$ is of full rank, we show that the image of the algebra homomorphisms
is in the corresponding quantum cluster algebra.
