  We provide the localization procedure for monoidal categories by a real
commuting family of braiders. For an element $w$ of the Weyl group,
$\mathscr{C}_w$ is a subcategory of modules over quiver Hecke algebra which
categorifies the quantum unipotent coordinate algebra $A_q[\mathfrak{n}(w)]$.
We construct the localization $\widetilde{\mathscr{C}_w}$ of $\mathscr{C}_w$ by
adding the inverses of simple modules which correspond to the frozen variables
in the quantum cluster algebra $A_q[\mathfrak{n}(w)]$. The localization
$\widetilde{\mathscr{C}_w}$ is left rigid and we expect that it is rigid.
