Let $(T, \mu , \eta )$ a monad on the category $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle: \mathscr{C}^T \to \mathscr{C}$ where $\mathscr{C}^T$ is the category of $T$-agebras, objects are couples $(A, a)$ where $a: T(A) \to A$ such that $a \circ \eta_A=1$, $a \circ \mu= a \circ T(a) $.

The above adjunction induce on $\mathscr{C}$ the some triple $T$, **but** on $\mathscr{C}^T$ there is a cotriple $(cT, \delta, \epsilon )$ with $cT(A, a)= (T(A), \mu_A)$, $\epsilon_{(A, a)}: cT(A, a) \to (A, a)$ is give by $a: T(A)\to A$, $\delta: cT(A, a) \to cT\circ cT(A, a) $ i.e. $\delta: (T(A), \mu_A) \to (TT(A), \mu_{TA})) $ is give by $T\eta_A: T(A)\to TTA$.
Now a coalgebra of $cT$ is a couple $((A, a), a^1)$ where $(A, a)$ is a $T$-algebra and $a'$ is a $T$-algebra morphism $a': (A, a) \to (T(A), \mu_A)$ such that $a'\circ \epsilon_{A, a}=1 $ and $T(\eta_A)\circ a'= T(a')\circ a' $.

If $T$ come from the free groups adjunction $\langle F, U, \epsilon , \eta \rangle: Gr \to Set$ the condition $a'\circ \epsilon_{A, a}=1 $ makes me suspect that the only cotriples $((A, a), a^1)$ are free-triples (i.e. free groups).

But in the work of J. De Vries (Lecture Notes Math N.540 p.654) in Th 2.3 (p. 659) he show a monadic functor $TTG \to TopGr \times Top$

(where $TTG$ is the topological transformation groups category, $TopGr$ the topological groups category, $Top$ topological spaces category)

where the algebras of the associate monads are the topological actions $\pi: G\times X \to X$.

In Th.2.6 he considered the comonads on the algebras category as in mine initial presentation, and prove that the associated coalgebras are topological actions $\pi: G\times X \to X$ with a continuous cross section $(S, u)$.

So it seems to me that the problem is not trivial, although it come from very simple considerations, however I have not found anywhere.

We can iterate this process of iteration, what we get? What about (co)triple (co)homology? ecc.ecc.

**I ask**: Do you know where (if exist) I can find a study of this issue?

If not, I hope it can be useful inspiration for those who have shoulders most powerful than mine.