Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms \documentclass[12pt]{minimal}
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\begin{document}$${\phi, \psi: C\to A}$$\end{document} are approximately unitarily equivalent if and only if\documentclass[12pt]{minimal}
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\begin{document}$$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$$\end{document}where T(A) is the tracial state space of A. In this paper we prove the following: Given \documentclass[12pt]{minimal}
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\begin{document}$${\kappa\in KL(C,A)}$$\end{document} with \documentclass[12pt]{minimal}
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\begin{document}$${\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$\end{document} and with κ([1C]) = [1A] and a continuous affine map \documentclass[12pt]{minimal}
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\begin{document}$${\lambda: T(A)\to T_{\mathfrak f}(C)}$$\end{document} which is compatible with κ, where \documentclass[12pt]{minimal}
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\begin{document}$${T_{\mathfrak f}(C)}$$\end{document} is the convex set of all faithful tracial states, there exists a unital monomorphism \documentclass[12pt]{minimal}
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\begin{document}$${\phi: C\to A}$$\end{document} such that\documentclass[12pt]{minimal}
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\begin{document}$$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$$\end{document}for all \documentclass[12pt]{minimal}
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\begin{document}$${c\in C_{s.a.}}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${\tau\in T(A).}$$\end{document} Denote by \documentclass[12pt]{minimal}
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\begin{document}$${{\rm Mon}_{au}^e(C,A)}$$\end{document} the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$$\end{document}where KLT(C, A)++ is the set of compatible pairs of elements in KL(C, A)++ and continuous affine maps from T(A) to \documentclass[12pt]{minimal}
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\begin{document}$${T_{\mathfrak f}(C).}$$\end{document} Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and \documentclass[12pt]{minimal}
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\begin{document}$${\kappa\in KL(C(X), A)}$$\end{document} with \documentclass[12pt]{minimal}
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\begin{document}$${\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$\end{document} for which there is no homomorphism h: C(X) → A so that [h] = κ.
