Classification of homomorphisms from $C(\mathrm{\Omega })$ to a $C^{\ast }$-algebra

Let $\mathrm{\Omega }$ be a compact subset of $\mathbb{C}$ and let $A$ be a unital simple, separable $C^{\ast }$-algebra with stable rank one, real rank zero and strict comparison. We show that, given a Cu-morphism $\alpha :\mathrm{C}\mathrm{u}(C(\mathrm{\Omega }))\to \mathrm{C}\mathrm{u}(A)$ with $\alpha (⟨\text{mathds}{1}_{\mathrm{\Omega }}⟩)\le ⟨1_A⟩$, there exists a homomorphism $\phi :C(\mathrm{\Omega })\to A$ such that $\mathrm{C}\mathrm{u}(\phi )=\alpha$ and $\phi$ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^{\ast }$-algebras to $A$ in terms of the Cuntz semigroup.