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If an endofunctor $F$ has a right adjoint $G$ and $(F,\eta ,\mu )$ is a monad, the mate correspondence turns $\eta :1\to F$ and $\mu :F^2\to F$ into a counit $G\to 1$ and comultiplication $G\to G^2$. The reversed mate correspondence turns the monad laws into the comonad laws and identifies $F$-algebras with $G$-coalgebras.

An adjunction $F⊣G$ is equivalently specified by a unit and counit of an adjunctionsatisfying the triangle identities

Let $F_i⊣G_i$ have units ${\eta }_i$ and counits ${\epsilon }_i$. The mate correspondence sends $\alpha :F_1\to F_2$ toConversely, $\beta :G_2\to G_1$ givesNaturality and the triangle identities show that the two constructions are inverse, yielding the required bijection of natural transformations.

Now let the endofunctor $F$ carry a monad $(F,\eta ,\mu )$ and let $F⊣G$. Taking right mates turnsinto a counit $ϵ:G\to 1$ and comultiplication $\delta :G\to G^2$. Since mates reverse composition, the monad unit and associativity laws become the comonad counit and coassociativity laws. Moreover, a map $FA\to A$ corresponds under the adjunction to a map $A\to GA$, and the algebra axioms correspond exactly to the coalgebra axioms. This is the monad on a left adjoint induces a comonad on its right adjoint construction and gives an isomorphism of the two structure categories.

For a monoid $M$, the free-$M$-set functor is $F(X)=M\times X$, and its algebras are precisely left $M$-sets, the objects of $[M,\mathbf{Set}]$. Its right adjoint is $G(X)=X^M$. The preceding isomorphism identifies $[M,\mathbf{Set}]$ with the Eilenberg-Moore category of coalgebras for the induced comonad on sets, compatibly with the forgetful functor. Therefore