Let be a functor and an object. The essential fiber of over is the following category:
The essential fiber can be identified with the pseudopullback of along the functor from the terminal category which picks out the object . It can also be identified with a homotopy fiber in the canonical model structure on Cat. When groupoids are identified with homotopy 1-types, the essential fiber actually coincides with the classical homotopy fiber (up to equivalence).
On the other hand, when is a Street fibration (the non-evil version of a Grothendieck fibration), then essential fibers do not coincide with strict fibers, and essential fibers are the more useful notion. In particular, the correspondence between fibrations and pseudofunctors only goes through for Street fibrations if one defines the pseudofunctor using essential fibers.