  Let $R$ be a quiver Hecke algebra, and let $\mathcal{C}_{w,v}$ be the
category of finite-dimensional graded $R$-module categorifying a
$q$-deformation of the doubly-invariant algebra $^{N'(w)} \mathbb{C}[N] ^{N(v)}
$. In this paper, we prove that the localization $\tilde{\mathcal{C}}_{w,v}$ of
the category $\mathcal{C}_{w,v}$ can be obtained as the localization by right
braiders arising from determinantial modules. As its application, we show
several interesting properties of the localized category
$\tilde{\mathcal{C}}_{w,v} $ including the right rigidity.
