  It is shown that if $A$ and $B$ are unital separable simple nuclear $\mathcal
Z$-stable C$^*$-algebras and there is a unital embedding $A \rightarrow B$
which is invertible on $KK$-theory and traces, then $A \cong B$. In particular,
two unital separable simple nuclear $\mathcal Z$-stable C$^*$-algebras which
either have real rank zero or unique trace are isomorphic if and only if they
are homotopy equivalent. It is further shown that two finite strongly
self-absorbing C$^*$-algebras are isomorphic if and only if they are
$KK$-equivalent in a unit-preserving way.
