In general the μ is not a real number, but a linear isomorphism between the two utilities, invariantly defined by some process.
“Isomorphism” (and “between”) seems like a very wrong word to use here. Linear combination of two utilities, perhaps.
I suspect you misunderstand. The two isomorphic utilities (i.e. utility functions) are U2 and μU2. You seem to be referring to the linear combination of U1 and U2.
I suspect you misunderstand. The two isomorphic utilities (i.e. utility functions) are U2 and μU2. You seem to be referring to the linear combination of U1 and U2.
I’ve added an addendum to the post, laying out what μ actually is.
Though the whole addendum could be summarised as: yes, μ is pretty much what you’d expect. :-)