Contravariant functoriality of E^∞ for open inclusions

Ascertain whether the local homology theory E^∞: UBC → 𝔻 has contravariant functoriality with respect to inclusions of open subsets, analogous to the contravariant functoriality known for the locally finite theory ΣE(*)^{lf}(-).

Background

Within the framework of coarse and local homology theories, the authors construct two related functors: the locally finite homology theory ΣE(*)^{lf}(-), which possesses additional contravariant functoriality for inclusions of open subsets, and the local homology theory E^∞ derived from a strong coarse homology theory via the cone construction.

They highlight uncertainty about whether E^∞ enjoys the same contravariant functoriality as ΣE(*)^{lf}(-). Resolving this would clarify the naturality properties of E^∞ and potentially broaden its applicability in Mayer–Vietoris and excision contexts.