= Solution
Use the orientation $F:\mathcal C\to\mathcal D$ and $G:\mathcal D\to\mathcal C$. An <adjunction> $F\dashv G$ is equivalently specified by the <natural transformations>
$$
\eta:1_{\mathcal C}\Rightarrow GF,\qquad
\varepsilon:FG\Rightarrow1_{\mathcal D},
$$
called the <unit and counit of an adjunction>, satisfying the <triangle identities for an adjunction>
$$
\boxed{\varepsilon_{FC}\circ F(\eta_C)=1_{FC},\qquad
G(\varepsilon_D)\circ\eta_{GD}=1_{GD}.}
$$
The corresponding <natural bijection> is $\mathcal D(FC,D)\cong\mathcal C(C,GD)$, with $f\mapsto G(f)\eta_C$ and inverse $g\mapsto\varepsilon_DF(g)$. \b[The two triangular equations are the required compatibility conditions.] No proof of equivalence of the formulations is needed here.
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