= Solution
Write the <adjunction> as the <natural bijection>
$$
\Phi_{A,B}:\mathcal D(FA,B)\longrightarrow\mathcal C(A,GB).
$$
The <unit of an adjunction> and <counit of an adjunction> are the <natural transformations> whose components are
$$
\boxed{\eta_A=\Phi_{A,FA}(1_{FA}),\qquad
\varepsilon_B=\Phi^{-1}_{GB,B}(1_{GB}).}
$$
<Naturality> of the <hom-set> bijections gives, for $f:FA\to B$ and $g:A\to GB$,
$$
\Phi_{A,B}(f)=G(f)\eta_A,\qquad
\Phi^{-1}_{A,B}(g)=\varepsilon_B F(g).
$$
For example, postcomposition by $f$ in the first <hom-set> corresponds to postcomposition by $Gf$ in the second, giving the first formula; naturality in $A$ gives the second. The same <naturality> says that, for $a:A\to A'$ and $b:B\to B'$,
$$
GF(a)\eta_A=\eta_{A'}a,\qquad b\varepsilon_B=\varepsilon_{B'}FG(b).
$$
Thus these components do define the asserted <natural transformations> $1_{\mathcal C}\to GF$ and $FG\to1_{\mathcal D}$.
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