 Splitting a one-triangle adjunction idempotent

The one-triangle adjunction idempotent splits if and only if $G$ has a left adjoint. From a splitting $F\stackrel{r}{\to }H\stackrel{s}{\to }F$, define the new unit $Gr\eta$ and counit $\epsilon s_G$; the absorption identities imply both triangle identities for an adjunction. Conversely, for $H⊣G$ with unit $\psi :1\Rightarrow GH$ and counit $\phi :HG\Rightarrow 1$, the splitting maps are $r={\epsilon }_{H}F\psi$ and $s={\phi }_{F}H\eta$. Their composites are $rs=1_H$ and $sr=e$.