adjoint triple  This entry is about the notion of adjoint triple
involving 3 functors. Please do not mix with the notion of
adjoint monads, which were also sometimes called adjoint
triples, with “triple” then being a synonym for monad. However,
an adjoint triple in the sense here does induce an adjoint
monad!     An adjoint triple of functors  (FGH):CD ( F\dashv
G\dashv H) : C \to D  is a triple of functors F,H:CDF,H \colon C
\to D and G:DCG \colon D \to C together with adjunction data
FGF\dashv G and GHG\dashv H.    The two adjunctions imply of
course that GG preserves all limits and colimits that exist in
DD.   Every adjoint triple   gives rise to an adjoint pair
(GFGH):CC (G F \dashv G H) : C \to C  consisting of a monad GFG
F left adjoint to the comonad GHG H on CC;  as well as an
adjoint pair  (FGHG):DD. ( F G \dashv H G ) : D \to D \,.  See
adjoint monad for more.  In general there is a duality (an
antiequivalence of categories) between the category of monads
having right adjoints and comonads having l