$[\![ A, B ]\!] _{\mathrm{cp}}^G$ is the set of homotopy classes of completely positive $G$-equivariant asymptotic morphisms from $A$ to $B$. Moreover the Kasparov product is given by the composition of asymptotic morphisms. Hence $\psi$ gives an element ${\mathrm{KK}}^G(A \otimes C(X),A)$ such that $[\psi] \circ \iota ={\mathrm{id}}_A$. As an immediate consequence, we obtain the following.

\[thm:Rokinj\] Let $A$ be a separable unital $G$-${\mathrm{C}^*}$-algebra. If $A$ has continuous Rokhlin property, then $A$ is ${\mathfrak{J}}_G$-injective. Moreover, if $A$ has continuous Rokhlin dimension with commuting tower less than $d-1$, then $A$ is ${\mathfrak{J}}^{d}_G$-injective.

Here, we say that a separable $G$-${\mathrm{C}^*}$-algebra $A$ is ${\mathfrak{J}}_G^d$-injective if the ${\mathrm{KK}}^G$-morphism induced from the canonical inclusion $A \to A \otimes C(E_dG)$ has a left inverse (see [@mathOA150806815] for more details).

\[cor:equiv\] Let $A$ and $B$ be $G$-${\mathrm{C}^*}$-algebras with finite (continuous) Rokhlin dimension with commuting tower. For $\phi \in {\mathrm{KK}}^G(A, B)$, $\phi$ is a ${\mathrm{KK}}^G$-equivalence if and only if $\operatorname{Res}_G \phi$ is a ${\mathrm{KK}}$-equivalence. Moreover, if $A$ and $B$ are in the UCT class, $\phi$ is a ${\mathrm{KK}}^G$-equivalence if and only if $(\operatorname{Res}_G\phi) _*:{\mathrm{K}}_*(A) \to {\mathrm{K}}_*(B)$ is an isomorphism.

It follows from Theorem \[thm:Rokinj\], Proposition \[prp:RokcRok\], Lemma \[lem:avbdl\], Theorem 3.4 of [@mathOA150806815] and the universal coefficient theorem (Proposition 7.3 