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