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Let $A$ be a separable $C^*$-algebra. We prove that its stabilized secondsuspension $S^2A\otimes \mathcal K$ and the $C^*$-algebra $qA\otimes \mathcalK$ constructed by Cuntz in the framework of his picture of KK-theory areasymptotically equivalent. This means that there exist asymptotic morphismsfrom each to the other whose compositions are homotopic to the identity maps.This result yields an easy description of the natural transformation fromKK-theory to E-theory. One more corollary is the following. T. Loring ([3])proved that any asymptotic morphism from $\qC$ to any $C^*$-algebra $B$ ishomotopic to a $\ast$-homomorphism. We prove that the same is true when $\C$ isreplaced by any nuclear $C^*$-algebra $A$ and when $B$ is stable.

The C*-algebras qA\otimes K and S^2A\otimes K are asymptotically equivalent