  Let $U'_q(\mathfrak{g})$ be a twisted affine quantum group of type
$A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $\mathfrak{g}_{0}$ be the
finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin
quiver of type $\mathfrak{g}_{0}$, we define a full subcategory ${\mathcal
C}_{Q}^{(2)}$ of the category of finite-dimensional integrable
$U'_q(\mathfrak{g})$-modules, a twisted version of the category ${\mathcal
C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of
affine Schur-Weyl duality, we construct an exact faithful KLR-type duality
functor ${\mathcal F}_{Q}^{(2)}: Rep(R) \rightarrow {\mathcal C}_{Q}^{(2)}$,
where $Rep(R)$ is the category of finite-dimensional modules over the quiver
Hecke algebra $R$ of type $\mathfrak{g}_{0}$ with nilpotent actions of the
generators $x_k$. We show that ${\mathcal F}_{Q}^{(2)}$ sends any simple object
to a simple object and induces a ring isomorphism $K(Rep(R)) \simeq K({\mathcal
C}_{Q}^{(2)})$.
