Problem on unit and co-unit in category theory. We assume that $F:C\rightarrow D$ is a left adjoint to $G:D\rightarrow C$ and $\alpha:FG\rightarrow id_D$, $\beta:id_C\rightarrow GF$ are the co-unit and the unit. 
If $T:C\rightarrow C$ is the functor $T=GF$, then $G\circ \alpha \circ F$ defines a morphism of functors $\mu:T\circ T\rightarrow T$.
(i) How to prove that unit $\beta$ is indeed a unit for $\mu$, in the sense that the composites $id_T\circ \beta:T\rightarrow T\circ T\rightarrow T$ and $\beta \circ id_T:T\rightarrow T\circ T\rightarrow T$ are the identity?
(ii) What will be the corresponding construction for the functor $U=GF$?
Thank you in anticipation. 

You seem to have confusions in notation, e.g. $id_T\circ\beta$ is horizontal composition, and not the composition of mappings $\beta$ and $id_T$. Let me write $T\beta$ instead. (Aactually it maps an object $c\in C$ to the arrow $T(\beta(c))$.)
(i) We need the equations $\alpha F \circ F\beta=1_F$ and $G\alpha \circ \beta G=1_G$:
$$(\mu\circ T\beta)=G\alpha F\circ GF\beta =G(\alpha F\circ F\beta)=G\,1_F=1_{GF}=1_T$$
and similarly the other one.
(ii) I guess you meant $U=FG:D\to D$. This determines a comonad with $\eta:U\to UU$ which satisfies coassociativity, with counit $\alpha$.
