  We continue the study of realization of the prefundamental modules
$L_{r,a}^{\pm}$, introduced by Hernandez and Jimbo, in terms of unipotent
quantum coordinate rings as in [J-Kwon-Park, Int. Math. Res. Not., 2023]. We
show that the ordinary character of $L_{r,a}^{\pm}$ is equal to that of the
unipotent quantum coordinate ring $U_q^-(w_r)$ associated to fundamental $r$-th
coweight. When $r$ is cominuscule, we prove that there exists a
$U_q(\mathfrak{b})$-module structure on $U_q^-(w_r)$, which is isomorphic to
$L_{r,a\eta_r}^\pm$ for some $\eta_r \in \mathbb{C}^\times$.
