 Solution

For $c\in \mathcal{C}$ and $d\in \mathcal{D}$, define maps of hom-setsThey are natural in $c,d$, using naturality of $\eta ,\epsilon$. The one supplied triangle identity for an adjunction givesNaturality of $\epsilon$ also givesSince $\alpha \beta =1$, the composite $\beta \alpha$ is idempotent. Evaluating its equation $(\beta \alpha {)}^2=\beta \alpha$ on $f=1_{Fc}$ gives $e_c^2=e_c$. As $e$ is a composite of natural transformations, this proves that $e$ is an idempotent in the functor category, including the required naturality.

Suppose this one-triangle adjunction idempotent splits as natural transformations $F\stackrel{r}{\to }H\stackrel{i}{\to }F$ with $ri=1_H$ and $ir=e$. The maps $h\mapsto hr$ and $f\mapsto fi$ give inverse bijections between $\mathcal{D}(Hc,d)$ and the subset of $\mathcal{D}(Fc,d)$ consisting of maps with $f{e}_c=f$. Indeed $(hr)e_c=hrir=hr$, $(hr)i=h$, and $(fi)r=f{e}_c=f$. The maps $\alpha ,\beta$ give inverse bijections between that subset and $\mathcal{C}(c,Gd)$, since $\beta \alpha (f)=f{e}_c$ and $\alpha \beta =1$. Combining them gives the natural bijectionwith inverse $g\mapsto {\epsilon }_{d}F(g)i_c$. Hence $H⊣G$. Its adjunction unit is $Gr\eta$ and its adjunction counit is $\epsilon i_G$.

Conversely, suppose $H⊣G$, with natural bijection $\theta :\mathcal{D}(Hc,d)\cong \mathcal{C}(c,Gd)$. Set $\tau ={\theta }^{-1}\alpha$ and $\sigma =\beta \theta$. Then $\tau \sigma =1$ and $\sigma \tau (f)=f{e}_c$. By the covariant Yoneda lemma, the transformation $\tau :\mathcal{D}(Fc,-)\to \mathcal{D}(Hc,-)$ is precomposition with a unique morphism $i_c:Hc\to Fc$, and $\sigma$ is precomposition with a unique morphism $r_c:Fc\to Hc$. The equation $\tau \sigma =1$ evaluated at $1_{Hc}$ gives $r_c{i}_c=1_{Hc}$; the equation $\sigma \tau (f)=f{e}_c$ evaluated at $1_{Fc}$ gives $i_c{r}_c=e_c$. Naturality in $c$ of $\tau ,\sigma$ implies, again by the same Yoneda lemma, that $i,r$ are natural transformations. Thus they split $e$ in the whole functor category, rather than merely at individual objects. We have proved