  Let $A$, $B$ be separable C*-algebras, $B$ stable. Elements of the E-theory
group $E(A,B)$ are represented by asymptotic homomorphisms from the second
suspension of $A$ to $B$. Our aim is to represent these elements by (families
of) maps from $A$ itself to $B$. We have to pay for that by allowing these maps
to be even further from $*$-homomorphisms. We prove that $E(A,B)$ can be
represented by pairs $(\varphi^+,\varphi^-)$ of maps from $A$ to $B$ that are
not necessarily asymptotic homomorphisms, but have the same deficiency from
being ones. Not surprisingly, such pairs of maps can be viewed as pairs of
asymptotic homomorphisms from some C*-algebra $C$ that surjects onto $A$, and
the two maps in a pair should agree on the kernel of this surjection. We give
examples of full surjections $C\to A$, i.e. those, for which all classes in
$E(A,B)$ can be obtained from pairs of asymptotic homomorphisms from $C$.
