E-theory is compactly assembled

We show that the equivariant E-theory category EₛₑₚG for separable C*-algebras is a compactly assembled stable ∞-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 EₛₑₚG. As an application we investigate a topological enrichment of the homotopy category of a compactly assembled ∞-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.