  Let ${\mathbf U}^-_q$ be the negative half of the quantum group associated to
a Kac-Moody algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the
quantum group obtained by a folding of ${\mathfrak g}$. Let ${\mathbf A} =
{\mathbf Z}[q,q^{-1}]$. McNamara showed that $\underline{\mathbf U}^-_q$ is
categorified over a certain extenion ring $\widetilde{\mathbf A}$ of ${\mathbf
A}$, by uing the folding theory of KLR algebras. He posed a question whether
$\widetilde{\mathbf A}$ coincides with ${\mathbf A}$ or not. In this paper, we
give an affirmative answer for this problem.
