Right adjoint of a monad is a comonad

Question

I'm having trouble proving the following statement:

If a monad $T$ has a right adjoint $K$, then $K$ is a comonad and the categories of $T$-algebras and $K$-coalgebras are isomorphic.

So far I've attempted to show using the counit of the adjunction $\epsilon: TK\to \operatorname{Id}_{\mathscr C} $ and the unit of the monad $\theta: \operatorname{Id}_{\mathscr C}\to T$ to try and see that $(K,K\epsilon \theta K, \epsilon \theta K)$ defines the comonad, but so far I've been unable to do so.

Answer 1

So apparently i was so distracted i completely missed i was taking the comultiplication to be a morphism $K^2\to K$ instead of $K\to K^2$, which was an evident first mistake. Turns out there is a nice abstract approach using the theory of mates, see this paper by aaron lauda.