= Solution
An <adjunction> $F\dashv G$ is a family of <bijections>
$$
\Phi_{A,B}:\mathcal D(FA,B)\xrightarrow{\sim}\mathcal C(A,GB)
$$
natural in both variables. Its <adjunction unit> and <adjunction counit> are $\eta_A=\Phi_{A,FA}(1_{FA})$ and $\varepsilon_B=\Phi^{-1}_{GB,B}(1_{GB})$. Naturality of the correspondence gives
$$
\Phi(u)=Gu\,\eta_A,\qquad \Phi^{-1}(v)=\varepsilon_BFv.
$$
For $f:A\to A'$, naturality in both variables evaluates $\Phi(Ff)$ in two ways, giving $GFf\,\eta_A=\eta_{A'}f$. Thus $\eta$ is a <natural transformation>; the dual calculation gives naturality of $\varepsilon$.
Applying the inverse correspondence to $\eta_A$ and the correspondence to $\varepsilon_B$ yields the <triangle identities for an adjunction>:
$$
\boxed{\varepsilon_{FA}F\eta_A=1_{FA},\qquad G\varepsilon_B\eta_{GB}=1_{GB}.}
$$
They express that transposing an <identity morphism> and transposing back returns that identity.
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