= Solution
Define the correspondence from the given <natural transformations> by $u\mapsto Gu\,\eta_A$, with candidate inverse $v\mapsto\varepsilon_BFv$. For $u:FA\to B$, naturality of $\varepsilon$ and a <triangle identity for an adjunction> give
$$
\varepsilon_BFG(u)F\eta_A=u\varepsilon_{FA}F\eta_A=u.
$$
For $v:A\to GB$, naturality of $\eta$ and the other triangle identity give
$$
G\varepsilon_BGF(v)\eta_A=G\varepsilon_B\eta_{GB}v=v.
$$
These are inverse <bijections>. Naturality of $\eta$, $\varepsilon$, $F$ and $G$ makes the <bijections> natural in $A$ and $B$, so they define an adjunction with the required unit and counit. Uniqueness follows from the formulas in the previous part: any adjunction with that unit and counit must have exactly these transposition maps.
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