  The behaviour of limits of weak morphisms in 2-dimensional universal algebra
is not 2-categorical in that, to fully express the behaviour that occurs, one
needs to be able to quantify over strict morphisms amongst the weaker kinds.
F-categories were introduced to express this interplay between strict and weak
morphisms. We express doctrinal adjunction as an F-categorical lifting property
and use this to give monadicity theorems, expressed using the language of
F-categories, that cover each weaker kind of morphism.
