  For C*-algebras $A$ and $B$, we generalize the notion of a quasihomomorphism
from $A$ to $B$, due to Cuntz, by considering quasihomomorphisms from some
C*-algebra $C$ to $B$ such that $C$ surjects onto $A$, and the two maps forming
a quasihomomorphism agree on the kernel of this surjection. Under an additional
assumption, the group of homotopy classes of such generalized
quasihomomorphisms coincides with $KK(A,B)$. This makes the definition of
Kasparov's bifunctor slightly more symmetric and gives more flexibility for
constructing elements of $KK$-groups. These generalized quasihomomorphisms can
be viewed as pairs of maps directly from $A$ (instead of various $C$'s), but
these maps need not be $*$-homomorphisms.
