Using the Dadarlat isomorphism, we give a characterization for the
Kasparov product of $C^*$-algebra extensions. A certain relation
between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when
$B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and
$KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general.
extension, Kasparov product, $KK$-group
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]