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Let X be a compact metric space and A = C(X). Suppose that B is a class of unital C[sup *]-algebras satisfying certain conditions, we prove the following: For any ∈ > O, finite set F ⊂ A, there is an integer I such that if φ, ψ : A → B(B ∈ B) are sufficiently multiplicative morphisms (e.g. when both φ and ψ are *-homomorphisms) which induce same K-theoretical maps, then there are a unitary u ∈ M[sub l+1](B) and a homomorphism σ : A → M[sub l](B) with finite dimensional image such that υ[sup *] diag(φ(ƒ),σ(ƒ))υ- diag(ψ(ƒ), σ(ƒ)) < ∈ for all &fnof ∈ F. In particular, the integer I does not depend on B, φ and ψ. This feature has important applications to the classification theory of nuclear C[sup *]-algebras. [ABSTRACT FROM AUTHOR]