Characterization of the unit object in localized quantum unipotent category

For the quiver Hecke algebra $R$, let $R\text{-gmod}$ be the category of finite-dimensional graded $R$-modules, and let $\widetilde{R\text{-gmod}[w]}$ be the localization of $R\text{-gmod}$. Kashiwara and the second author showed the set of equivalence classes of simple objects up to grading shifts $\mathrm{I}\mathrm{r}\mathrm{r}(\widetilde{R\text{-gmod}[w]})$ in $\widetilde{R\text{-gmod}[w]}$ has a crystal structure, and $\mathrm{I}\mathrm{r}\mathrm{r}(\widetilde{R\text{-gmod}[w]})$ is isomorphic to the so-called cellular crystal ${\mathbb{B}}_{\mathbf{i}}$. This isomorphism induces a function ${\epsilon }_i^{\ast }$ on ${\mathbb{B}}_{\mathbf{i}}$. We give an explicit formula of ${\epsilon }_i^{\ast }$, and using this formula, we give a characterization of the unit object of $\widetilde{R\text{-gmod}[w]}$ for the case of classical finite types.