Suppose $F:C\rightarrow D$ and $ G:D\rightarrow C$ are functors, $\eta :1_{C}\rightarrow GF$ and $\varepsilon :FG\rightarrow 1_{D}$ are natural transformations. Suppose further that $G\varepsilon \cdot \eta G=1_{G}$. I want to prove that $\varepsilon F\cdot F\eta $ is an idempotent and that $G$ has a left adjoint if and only if it splits. I think it will be pretty easy to show that if $G$ has a left adjoint $H$, then $\varepsilon F\cdot F\eta $ splits as $F\overset{\alpha }{\rightarrow}H\overset{\beta }{\rightarrow}F$ and conversely, if $\varepsilon F\cdot F\eta $ splits then $H$ is a left adjoint.
But I am having trouble with the very first assertion: that $\varepsilon F\cdot F\eta $ is an idempotent. I want to use naturality and the fact that $G\varepsilon \cdot \eta G=1_{G}$ but I keep getting squares that do not commute. Can someone point me in the right direction?