An amnestic isofibration has the following lifting property: for any object in and any isomorphism in , there is a unique isomorphism such that . Indeed, if were any other isomorphism such that , then , so we must have .
If the composite is an amnestic functor, then is also amnestic.
Any strictly monadic functor is amnestic. Conversely, any monadic functor that is also an amnestic isofibration is necessarily strictly monadic.