  We provide a thorough algebraic analysis of three known completions having a
central role in the exact completions of Lawvere's doctrines: the one adding
comprehensive diagonals (i.e. forcing equality on terms to coincide with the
equality predicate), the one adding full comprehensions and the one adding
quotients. We show that all these 2-adjunctions are 2-monadic and that the
2-monads arising from these adjunctions are all property-like. This entails
that comprehensive diagonals, full comprehensions and quotients are algebraic
properties of an elementary doctrine. Finally, we discuss and present the
distributive laws between these 2-monads.
