Let $C$ be a general unital AH-algebra and let $A$ be a unital simple
$C^*$-algebra with tracial rank at most one. Suppose that $\phi, \psi: C\to A$
are two unital monomorphisms. We show that $\phi$ and $\psi$ are approximately
unitarily equivalent if and only if \beq\nonumber [\phi]&=&[\psi] {\rm in}
KL(C,A),\nonumber \phi_{\sharp}&=&\psi_{\sharp}\tand \phi^{\dag}&=&\psi^{\dag},
\eneq where $\phi_{\sharp}$ and $\psi_{\sharp}$ are continuous affine maps from
tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{\rm
f}(C)$ of $C$ induced by $\phi$ and $\psi,$ respectively, and $\phi^{\ddag}$
and $\psi^{\ddag}$ are induced homomorphisms from $K_1(C)$ into
$\Aff(T(A))/\overline{\rho_A(K_0(A))},$ where $\Aff(T(A))$ is the space of all
real affine continuous functions on $T(A)$ and $\overline{\rho_A(K_0(A))}$ is
the closure of the image of $K_0(A)$ in the affine space $\Aff(T(A)).$ In
particular, the above holds for $C=C(X),$ the algebra of continuous functions
on a compact metric space.

An approximate version of this is also obtained. We also show that, given a
triple of compatible elements $\kappa\in KL_e(C,A)^{++},$ an affine map
$\gamma: T(C)\to T_{\rm f}(C)$ and a \hm $\af: K_1(C)\to
\Aff(T(A))/\overline{\rho_A(K_0(A))},$ there exists a unital monomorphism
$\phi: C\to A$ such that $[h]=\kappa,$ $h_{\sharp}=\gamma$ and
$\phi^{\dag}=\af.$