If is a forgetful functor, then being an isofibration says that whatever stuff forgets can be “transported along isomorphisms.”
Isofibrations have a number of good properties. For example, any strict pullback of an isofibration is also a weak pullback. Any Grothendieck fibration or opfibration is an isofibration, but not conversely (unless is a groupoid).
The isofibrations are the fibrations in the canonical model structure on Cat. More generally, the fibrations in canonical model structures on various types of higher categories are usually some generalization of isofibrations. For example, the fibrations in the Lack model structure on 2-Cat have “equivalence lifting” and “local isomorphism lifting,” and the fibrations in the Joyal model structure for quasicategories have “equivalence lifting” at all levels.
This definition of isofibration is evil where it demands that ; if it only demanded , of course, any functor would qualify.