\beta_g \circ \varphi \circ \alpha_g^{-1} (x) \text{ as } n \to \infty$$ Then we have a $G$-equivariant $\ast$-homomorphism $\psi: A \to B$ which is approximately unitarily equivalent to $\varphi$. Moreover if $\varphi$ is an isomorphism, then $\psi$ can be taken to be an isomorphism.

Put $\theta: A \to B \otimes C(G)$ as $\theta(x)(g) := \beta_g \circ \varphi \circ \alpha_g^{-1}(x)$. Then $\theta$ is a $G$-equivariant $\ast$-homomorphism. Fix a $G$-invariant compact set $F \subset A$ and $\varepsilon > 0$. By assumption, we may take a unitary $u \in B \otimes C(G)$ such that $$\|u(\varphi(x) \otimes 1) u^* - \theta(x) \| < \varepsilon$$ for $x \in F$. Now we take the Rokhlin approximation $\chi: B \otimes C(G) \to B$ such that

-   $\| \chi(u)^* \chi(u) - 1\| < \varepsilon$, $\| \chi(u) \chi(u)^* - 1\| < \varepsilon,$

-   $\| \chi(u) \chi(\varphi(x)\otimes 1) \chi(u)^* - \chi \circ \theta(x) \| < \varepsilon$ for $x \in F$,

-   $\| \chi(\varphi(x) \otimes 1) - \varphi(x) \| < \varepsilon$ for $x \in F$.

Take the unitary $v := \chi(u) |\chi(u)|^{-1}$. Then we have $\| \chi(u) - v \| < \varepsilon/2$ and hence $$\| v \varphi(x) v^* - \chi \circ \theta(x) \| < \frac{3}{2}\varepsilon.$$ Since $\chi \circ \theta(x)$ is $G$-equivariant, 