= Solution
Write the <adjunction> bijections as $\Phi_{A,B}:\mathcal D(FA,B)\cong\mathcal C(A,GB)$, natural in $A,B$. The <adjunction unit> and <adjunction counit> are
$$
\boxed{\eta_A=\Phi_{A,FA}(1_{FA}),\qquad\varepsilon_B=\Phi_{GB,B}^{-1}(1_{GB}).}
$$
Thus $\eta:1_{\mathcal C}\Rightarrow GF$ and $\varepsilon:FG\Rightarrow1_{\mathcal D}$ are <natural transformations>, by naturality of the bijections. For any $h:FA\to B$ and $k:A\to GB$, their transposes satisfy
$$
\Phi(h)=Gh\,\eta_A,\qquad\Phi^{-1}(k)=\varepsilon_BFk.
$$
The first follows by naturality in the target applied to $1_{FA}$; the second follows by naturality in the source applied to $1_{GB}$. These formulas make the definitions operational for every arrow.
Back to article page
