  The extended $T$-systems are a number of short exact sequences in the
category of finite-dimensional modules over the quantum affine algebras of
types $A_n^{(1)}$ and $B_n^{(1)}$, introduced by Mukhin and Young as a
generalization of the $T$-systems. In this paper we establish the extended
$T$-systems for more general modules, which are constructed from an arbitrary
strong duality datum of type $A$. Our approach does not use the theory of
$q$-characters, and so also provides a new proof to the original Mukhin-Young's
extended $T$-systems.
