  Given an adjoint pair of functors $F,G$, the composite $GF$ naturally gets
the structure of a monad. The same monad may arise from many such adjoint pairs
of functors, however. Can one describe all of the adjunctions giving rise to a
given monad? In this paper we single out a class of adjunctions with especially
good properties, and we develop methods for computing all such adjunctions, up
to natural equivalence, which give rise to a given monad. To demonstrate these
methods, we explicitly compute the finitary homological presentations of the
free $A$-module monad on the category of sets, for $A$ a Dedekind domain. We
also prove a criterion, reminiscent of Beck's monadicity theorem, for when
there is essentially (in a precise sense) only a single adjunction that gives
rise to a given monad.
