  We prove that a finite family of commuting holomorphic self-maps of the unit
ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic
conjugacy to a family of commuting automorphisms of a possibly lower
dimensional ball, and that such conjugacy satisfies a universal property. As an
application we describe when a hyperbolic and a parabolic holomorphic self-map
of $\mathbb{B}^q$ can commute.
