  The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly
considered the conceptual basis for a full characterization of an electronic
system in its ground state by just the one-body particle density. In this
Part~II of a series of two articles, we aim at clarifying the status of this
theorem within different extensions of DFT including magnetic fields. We will
in particular discuss current-density-functional theory (CDFT) and review the
different formulations known in the literature, including the conventional
paramagnetic CDFT and some non-standard alternatives. For the former, it is
known that the Hohenberg-Kohn theorem is no longer valid due to
counterexamples. Nonetheless, paramagnetic CDFT has the mathematical framework
closest to standard DFT and, just like in standard DFT, non-differentiability
of the density functional can be mitigated through Moreau-Yosida
regularization. Interesting insights can be drawn from both
Maxwell-Schr\"odinger DFT and quantum-electrodynamical DFT, which are also
discussed here.
