Monad on a left adjoint induces a comonad on its right adjoint

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.