  We construct a monoidal category $\mathscr{C}_{w,v}$ which categorifies the
doubly-invariant algebra $^{N'(w)}\mathbb{C}[N]^{N(v)}$ associated with Weyl
group elements $w$ and $v$. It gives, after a localization, the coordinate
algebra $\mathbb{C}[\mathcal{R}_{w,v}]$ of the open Richardson variety
associated with $w$ and $v$. The category $\mathscr{C}_{w,v}$ is realized as a
subcategory of the graded module category of a quiver Hecke algebra $R$. When
$v= \mathrm{id}$, $\mathscr{C}_{w,v}$ is the same as the monoidal category
which provides a monoidal categorification of the quantum unipotent coordinate
algebra $A_q(\mathfrak{n}(w))_{\mathbb{Z}[q,q^{-1}]}$ given by
Kang-Kashiwara-Kim-Oh. We show that the category $\mathscr{C}_{w,v}$ contains
special determinantial modules $\mathsf{M}(w_{\le k}\Lambda, v_{\le k}\Lambda)$
for $k=1, \ldots, \ell(w)$, which commute with each other. When the quiver
Hecke algebra $R$ is symmetric, we find a formula of the degree of $R$-matrices
between the determinantial modules $\mathsf{M}(w_{\le k}\Lambda, v_{\le
k}\Lambda)$. When it is of finite $ADE$ type, we further prove that there is an
equivalence of categories between $\mathscr{C}_{w,v}$ and $\mathscr{C}_u$ for
$w,u,v \in \mathsf{W}$ with $w = vu$ and $\ell(w) = \ell(v) + \ell(u)$.
