  For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of
the convolution algebra of the equivariant K-group of a certain Steinberg type
graded quiver variety. We observe that it is affine quasi-hereditary and prove
that its category of finite-dimensional modules is identified with a block of
Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the
quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an
application, we show that Kang-Kashiwara-Kim's generalized quantum affine
Schur-Weyl duality functor gives an equivalence between the category of
finite-dimensional modules over the quiver Hecke algebra associated with $Q$
and Hernandez-Leclerc's category $\mathcal{C}_Q$, assuming the simpleness of
some poles of normalized R-matrices for type E.
