Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $phi, psi: Cto A$ are approximately unitarily equivalent if and only if $$ [phi]=[psi] {rm in} KL(C,A) and taucirc phi=taucirc psi tforal tauin T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $kappain KL(C,A)$ with $kappa(K_0(C)_+setminus {0})subset K_0(A)_+setminus {0}$ and with $kappa([1_C])=[1_A]$ and a continuous affine map $lambda: T(A)to T_{mathtt{f}}(C)$ which is compatible with $kappa,$ where $T_{mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $phi: Cto A$ such that $$ [phi]=kappaandeqn taucirc phi(c)=lambda(tau)(c) $$ for all $cin C_{s.a.}$ and $tauin T(A).$ Denote by ${rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$ Lambda: {rm Mon}_{au}^e (C,A)to KLT(C,A)^{++}, $$ where $KLT(C,A)^{++}$ is the set of compatible pairs of elements in $KL(C,A)^{++}$ and continuous affine maps from $T(A)$ to $T_{mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $kappain KL(C(X), A)$ with $kappa(K_0(C(X))_+setminus{0})subset K_0(A)_+setminus {0}$ for which there is no hm $h: C(X)to A$ so that $[h]=kappa.$
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