= Solution
For $c\in\mathcal C$ and $d\in\mathcal D$, define maps of <hom-sets>
$$
\alpha(f)=Gf\,\eta_c:\mathcal D(Fc,d)\to\mathcal C(c,Gd),\qquad
\beta(g)=\varepsilon_dFg:\mathcal C(c,Gd)\to\mathcal D(Fc,d).
$$
They are natural in $c,d$, using <naturality> of $\eta,\varepsilon$. The one supplied <triangle identity for an adjunction> gives
$$
\alpha\beta(g)=G\varepsilon_d\,GFg\,\eta_c=G\varepsilon_d\,\eta_{Gd}g=g.
$$
<Naturality> of $\varepsilon$ also gives
$$
\beta\alpha(f)=\varepsilon_dFGfF\eta_c=f\varepsilon_{Fc}F\eta_c=f e_c,
\qquad e_c=\varepsilon_{Fc}F\eta_c.
$$
Since $\alpha\beta=1$, 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 \b[$e$ is an idempotent in the <functor category>], including the required <naturality>.
Suppose this <one-triangle adjunction idempotent> splits as <natural transformations> $F\xrightarrow r H\xrightarrow i F$ 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 $fe_c=f$. Indeed $(hr)e_c=hrir=hr$, $(hr)i=h$, and $(fi)r=fe_c=f$. The maps $\alpha,\beta$ give inverse <bijections> between that <subset> and $\mathcal C(c,Gd)$, since $\beta\alpha(f)=fe_c$ and $\alpha\beta=1$. Combining them gives the <natural bijection>
$$
\mathcal D(Hc,d)\cong\mathcal C(c,Gd),\qquad h\longmapsto G(hr)\eta_c,
$$
with inverse $g\mapsto\varepsilon_dF(g)i_c$. Hence $H\dashv G$. Its <adjunction unit> is $Gr\,\eta$ and its <adjunction counit> is $\varepsilon\,i_G$.
Conversely, suppose $H\dashv G$, 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 $\sigma\tau(f)=fe_c$. 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 $r_ci_c=1_{Hc}$; the equation $\sigma\tau(f)=fe_c$ evaluated at $1_{Fc}$ gives $i_cr_c=e_c$. <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
$$
\boxed{G\text{ has a left adjoint}\iff e=\varepsilon_FF\eta\text{ splits in }[\mathcal C,\mathcal D].}
$$
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