  Recent studies have made key progress on the black hole/solitonic solutions
of the Einstein-Proca system. Firstly, fully non-linear dynamical evolutions of
the Kerr black hole superradiant instability, triggered by a Proca field, have
shown the formation of a new equilibrium state, a spinning black hole with
synchronised Proca hair. Secondly, non-linear evolutions of spinning Proca
stars have established that they are dynamically stable, unlike their scalar
cousins. Thirdy, separability of the Proca equation on the Kerr background has
been achieved. Motivated by these results, in this paper we reconsider Kerr
black holes with synchronised Proca hair. The separability of the Proca
equation on the Kerr background allows us to examine the stationary Proca
clouds in greater detail, in particular their dependence on the different
quantum numbers. These stationary clouds occur at a set of existence lines in
the Kerr parameter space, from which the black holes with synchronised Proca
hair bifurcate. We construct the domain of existence of these black holes,
comparing the fundamental states missed in the original study with the first
excited states and with the cousin scalar model, giving illustrative examples
of Kerr-like and non-Kerr-like BHs. In the vanishing event horizon limit, these
hairy black holes connect to the fundamental states of spinning Proca stars,
which include the dynamically stable solutions.
