  Let $A$ be a unital separable amenable \CA and $C$ be a unital \CA with
certain infinite property. We show that two full monomorphisms $h_1, h_2: A\to
C$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]$ in
$KL(A,C).$ Let $B$ be a non-unital but $\sigma$-unital \CA for which $M(B)/B$
has the certain infinite property. We prove that two full essential extensions
are approximately unitarily equivalent if and only if they induce the same
element in $KL(A, M(B)/B).$ The set of approximately unitarily equivalence
classes of full essential extensions forms a group. If $A$ satisfies the
Universal Coefficient Theorem, it is can be identified with $KL(A, M(B)/B).$
