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In our previous paper, we gave a presentation of the torus-equivariantquantum $K$-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 explicitideal. In this paper, we prove that quantum double Grothendieck polynomials,introduced by Lenart-Maeno, represent the corresponding (opposite) Schubertclasses in the quantum $K$-theory ring $QK_{H}(Fl_{n+1})$ under thispresentation. The main ingredient in our proof is an explicit formulaexpressing the semi-infinite Schubert class associated to the longest elementof the finite Weyl group, which is proved by making use of the generalChevalley formula for the torus-equivariant $K$-group of the semi-infinite flagmanifold associated to $SL_{n+1}(\mathbb{C})$.

A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part II: quantum double Grothendieck polynomials