there are $X$-equivariant $\ast$-homomorphisms $\phi_n \colon \mathfrak A_n \to \mathfrak B_{k_n}$ which generate a $\ast$-homomorphism $\phi \colon \mathfrak A \to \mathfrak B$. Then $\phi$ is $X$-equivariant.

Let $U \in \mathbb O(X)$ and $a\in \mathfrak A(U)$, so that we should show that $\phi(a) \in \mathfrak B(U)$. Given $\epsilon >0$ we may, by the $X$-equivariant structure of $\mathfrak A$, find an $N$ and an $a'$ in $\mathfrak A_N(U)$ such that $\iota_N(a') \approx_{\epsilon } a$. Thus $$\phi(a) \approx_{\epsilon} \phi ( \iota_N(a')) = \iota_{k_N}(\phi_N(a')) \in \mathfrak B(U),$$ since both $\iota_{k_N}$ and $\phi_N$ are $X$-equivariant. Thus $\phi(a)\in \mathfrak B(U)$.

Invariants for $C^{*}$-algebras over a totally disconnected space.
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Let $\mathcal{P} \subseteq {\ensuremath{\mathbb{N}}\xspace}$ be the set consisting of $0$ and all prime powers. The relevance of the set $\mathcal{P}$ in the Universal Multicoefficient Theorem is that the groups ${\ensuremath{\mathbb{Z}}\xspace}_{p} := {\ensuremath{\mathbb{Z}}\xspace}/p {\ensuremath{\mathbb{Z}}\xspace}$ for $p\in \mathcal{P}$ are exactly the indecomposable abelian groups.

For non-zero $p\in \mathcal{P}$, let $\mathbb{I}_p$ be the mapping cone of the unital $*$-homomorphism ${\ensuremath{\mathbb{C}}\xspace}\to \mathsf{M}_p({\ensuremath{\mathbb{C}}\xspace})$. For $p=0$, we let $\mathbb{I}_0 := {\ensuremath{\mathbb{C}}\xspace}$. It is convenient to denote $\mathbb{I}_p$ by $\mathbb{I}_p^0$ and its suspension $\Sigma\mathbb{I}_p$ by $\mathbb{I}_p^1$. Then for a $C^{*}$-algebra $\mathfrak{A}$: $$K_i(A;{\ensuremath{\mathbb{Z}}\xspace}_{p}) := {\ensuremath{\mathit{KK}}\xspace}_i(\mathbb{I}_p,A)\cong {\ensuremath{\mathit{KK}}\xspace}(\mathbb{I}_p^i,A), \quad i=0,1.$$

Let us set $\mathbb{I} := \bigoplus_{p\in \mathcal{P} 