  We explain that the Pontryagin product structure on the equivariant $K$-group
of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math.
{\bf 146} (2010)] coincides with the tensor structure on the equivariant
$K$-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke
Math. {\bf 169} (2020)]. Then, we construct an explicit isomorphism between the
equivariant $K$-group of a semi-infinite flag manifold with a suitably
localized equivariant quantum $K$-group of the corresponding flag manifold.
These exhibit a new framework to understand the ring structure of equivariant
quantum $K$-theory and the Peterson isomorphism.
