  We show that the equivariant $E$-theory category
$\mathrm{E}_{\mathrm{sep}}^{G}$ for separable $C^{*}$-algebras is a compactly
assembled stable $\infty$-category. We derive this result as a consequence of
the shape theory for $C^{*}$-algebras developed by Blackadar and Dardarlat and
a new construction of $\mathrm{E}_{\mathrm{sep}}^{G}$. As an application we
investigate a topological enrichment of the homotopy category of a compactly
assembled $\infty$-category in general and argue that the results of Carri\'on
and Schafhauser on the enrichment of the classical $E$-theory category can be
derived by specialization.
