I'm currently explaining monads
$$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$
to my brain and the "only" tricky thing are really the identity relations.
I came up with a potential argument why this definition is the natural thing to define, in particularly over easier candidates.
Is the following right?
In the case where, for all $A\in\mathrm{Ob}_{\bf C}$, we have
(edit)
$$\eta_{TA}\circ\mu_{A}=id_{TTA},$$
the monad constrcution (for programming purposes say) is useless because it actually just shifts all of the (type) structure, i.e. merely renaming all participants. The fact that $\mu$ is generally not the direct invers of $\eta$ allows there for something to be added or lost.
In the case $\mu$ is a natural isomorphism, the monad is said to be idempotent. In this case, the forgetful functor from the Eilenberg-Moore category of algebras is full and faithful.