= Solution
Write the <adjunction> as a <natural bijection> $\alpha_{C,D}:\mathcal D(FC,D)\to\mathcal C(C,GD)$. The <adjunction unit> and <adjunction counit> are defined by
$$
\eta_C=\alpha_{C,FC}(1_{FC}),\qquad\varepsilon_D=\alpha^{-1}_{GD,D}(1_{GD}).
$$
<Naturality> of $\alpha$ implies $GF(u)\eta_C=\eta_{C'}u$ for $u:C\to C'$ and $v\varepsilon_D=\varepsilon_{D'}FG(v)$ for $v:D\to D'$, so these are <natural transformations> $1_{\mathcal C}\to GF$ and $FG\to1_{\mathcal D}$. More explicitly, <naturality> at $f:FC\to D$ and $g:C\to GD$ gives the two transpose formulas
$$
\alpha(f)=G(f)\eta_C,\qquad\alpha^{-1}(g)=\varepsilon_DF(g).
$$
These formulas will also prove the <triangle identities for an adjunction>.
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