A monad $(T,\mu,\eta)$ is adjoint to a comonad $(G,\delta,\epsilon)$, if its underlying endofunctor $T$ is left adjoint to the underlying 1-cell $G$ of the comonad, and $\delta$ and $\epsilon$ are conjugate/adjoint/mate 2-cells to $\mu$ and $\eta$ in the sense explained below.
In fact given a monad $\mathbf{T} = (T,\mu^T,\eta^T)$ which has a right adjoint $G$, automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$.
Thus there is a bijective correspondence between monads having a right adjoint and comonads having a left adjoint (what Alexander Rosenberg calls duality). I am not sure that the terminology is optimal. In any case, it is a little more than a consequence of two general facts.
If $T\dashv G$ then $T^k \dashv G^k$ for every natural number $k$.
Given two adjunctions $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a bijection between the natural transformations $\phi:S'\Rightarrow S$ and natural transformations $\psi:T\Rightarrow T'$ such that
where the horizontal arrows are the natural bijections given by the adjunctions. Eilenberg and Moore would write $\phi\dashv\psi$ and talk about “adjointness for morphisms” (of functors), which is of course relative to the given adjunctions among functors. MacLane calls the correspondence conjugation (Categories for Working Mathematician, 99-102). It is a special case, of a general construction of mates, in terminology of Australian category school.
If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their unit and counit of course the upper arrow is $(S M\stackrel{f}\to N)\mapsto T f\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as
or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above diagram, as the horizontal arrows are invertible. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition $\psi = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e.
and checks that it works. The inverse is similarly given by the composition
This correspondence now enables in our special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit.
Every adjoint triple $F^*\dashv F_* \dashv F^!$ induces an adjoint pair $F_* F^*\dashv F_* F^!$. The endofunctor $F_* F^*$ is underlying a monad induced by the adjunction $F^*\dashv F_*$ and $F_* F^!$ is underlying a comonad induced by the adjuntion $F_*\dashv F^!$. This pair of a monad and a comonad are adjoint.
Given a small category $A$, the category $End(A)$ of endofunctors and natural transformations of endofunctors (with vertical composition as composition) is monoidal with respect to the composition as the tensor product of objects (endofunctor) and Godement product (horizontal composition) as the tensor product of morphisms (natural transformations). Hence we can consider operads and algebras over operads, as well as, dually, coalgebras over cooperads; or some other framework for general algebras and coalgebras (or even props).
In any case, given an adjunction $T\dashv G$, operations $T^n\to T$ dualize to cooperations $G\to G^n$, and more generally multioperations $T^k\to T^l$ dualize to the multioperations $G^l\to G^k$. We would like to sketch the proof that the identities for operations on $T$, correspond to the identities on cooperations on $G$ (and more generally there is a duality among the identities for multioperations). This is essentially the consequence of
Lemma. (Zoran) Given the adjunction $T\dashv G$ with unit $\eta$ and counit $\epsilon$, and the sequence
the composition $\alpha^*\circ\beta^*$ of the dual (in the above sense) sequence
equal to the dual $(\beta\circ\alpha)^*$ of $\beta\circ\alpha$,
Proof. Mike Shulman notices that this is a special case of known contravariant functoriality of mates, but here is a direct proof.
We need to prove that the composition
equals the composition
Note that in the two compositions there is an opposite order between the expressions involving $\alpha$ and those involving $\beta$. But anyway, their equality reduces to a naturality calculation (which in particular exchanges the order of $\alpha$ and $\beta$ in effect):
where $\rho := G^k \beta G^s \circ G^k \epsilon_l G^s = G^k\epsilon_l T^s G^s\circ G^k T^l G^l \beta G^s$. The commutativity of all small squares in the diagram is evident, except the lower left corner. This one follows by one of the triangle identities for the adjunction $T^l\dashv G^l$. Namely,
An adjoint modality is an example of a pair of adjoint monads.
There is a section 3, “adjoint triples” in
where triple is in the sense of monad. So we say instead a monad adjoint to a comonad. Distinguish from the adjoint triple of functors.
Discussion in the context of ambidextrous adjunctions is in