 One-triangle adjunction idempotent

Let $F:\mathcal{C}\to \mathcal{D}$, $G:\mathcal{D}\to \mathcal{C}$, $\eta :1\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1$ be natural transformations. If $G\epsilon {\eta }_G=1_G$, then $e={\epsilon }_{F}F\eta :F\Rightarrow F$ is an idempotent morphism in the functor category. Naturality implies $Ge\eta =\eta$ and $\epsilon e_G=\epsilon$; these absorption identities prove $e^2=e$. The omitted triangle measures exactly whether $e=1_F$.