Title: Categorical relations between Langlands dual quantum affine algebras: Doubly laced types,
Abstract: We prove that the Grothendieck rings of category $\mathcal{C}^{(t)}_Q$ over
quantum affine algebras $U_q'(\g^{(t)})$ $(t=1,2)$ associated to each Dynkin
quiver $Q$ of finite type $A_{2n-1}$ (resp. $D_{n+1}$) is isomorphic to one of
category $\mathcal{C}_{\mQ}$ over the Langlands dual $U_q'({^L}\g^{(2)})$ of
$U_q'(\g^{(2)})$ associated to any twisted adapted class $[\mQ]$ of $A_{2n-1}$
(resp. $D_{n+1}$). This results provide partial answers of conjectures of
Frenkel-Hernandez on Langlands duality for finite-dimensional representation of
quantum affine algebras.