  Let $\mathfrak{g}_0$ be a simple Lie algebra of type ADE and let
$U'_q(\mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We
show that there exists an action of the braid group $B(\mathfrak{g}_0)$ on the
quantum Grothendieck ring $K_t(\mathfrak{g})$ of Hernandez-Leclerc's category
$C_{\mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a
family of monoidal autofunctors $\{\mathscr{S}_i\}_{i\in \mathbb{Z}}$ on a
localization $T_N$ of the category of finite-dimensional graded modules over
the quiver Hecke algebra of type $A_{\infty}$. Under an isomorphism between the
Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring
$K_t({A^{(1)}_{N-1}})$, the functors $\{\mathscr{S}_i\}_{1\le i\le N-1}$
recover the action of the braid group $B(A_{N-1})$. We investigate further
properties of these functors.
