  We prove in full generality that the generalized quantum affine Schur-Weyl
duality functor, introduced by Kang-Kashiwara-Kim, gives an equivalence between
the category of finite-dimensional modules over a quiver Hecke algebra and a
certain full subcategory of finite-dimensional modules over a quantum affine
algebra which is a generalization of the Hernandez-Leclerc's category
$\mathcal{C}_Q$. This was previously proved in untwisted $ADE$ types by Fujita
using the geometry of quiver varieties, which is not applicable in general. Our
proof is purely algebraic, and so can be extended uniformly to general types.
