  In our previous paper, we gave a presentation of the torus-equivariant
quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ of the (full) flag manifold
$Fl_{n+1}$ of type $A_{n}$ as a quotient of a polynomial ring by an explicit
ideal. In this paper, we prove that quantum double Grothendieck polynomials,
introduced by Lenart-Maeno, represent the corresponding (opposite) Schubert
classes in the quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ under this
presentation. The main ingredient in our proof is an explicit formula
expressing the semi-infinite Schubert class associated to the longest element
of the finite Weyl group, which is proved by making use of the general
Chevalley formula for the torus-equivariant $K$-group of the semi-infinite flag
manifold associated to $SL_{n+1}(\mathbb{C})$.
