  We show that several apparently unrelated formulas involving left or right
Bousfield localizations in homotopy theory are induced by comparison maps
associated with pairs of adjoint functors. Such comparison maps are used in the
article to discuss the existence of functorial liftings of homotopical
localizations and cellularizations to categories of algebras over monads acting
on model categories, with emphasis on the cases of module spectra and algebras
over simplicial operads. Some of our results hold for algebras up to homotopy
as well; for example, if $T$ is the reduced monad associated with a simplicial
operad and $f$ is any map of pointed simplicial sets, then $f$-localization
coincides with $Tf$-localization on spaces underlying homotopy $T$-algebras,
and similarly for cellularizations.
