Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 18 / 4 / b / Solution

Define the correspondence from the given natural transformations by $u\mapsto Gu{\eta }_A$, with candidate inverse $v\mapsto {\epsilon }_{B}Fv$. For $u:FA\to B$, naturality of $\epsilon$ and a triangle identity for an adjunction giveFor $v:A\to GB$, naturality of $\eta$ and the other triangle identity giveThese are inverse bijections. Naturality of $\eta$, $\epsilon$, $F$ 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.