Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 20 / 6

Write the adjunction as a natural bijection ${\alpha }_{C,D}:\mathcal{D}(FC,D)\to \mathcal{C}(C,GD)$. The adjunction unit and adjunction counit are defined byNaturality of $\alpha$ implies $GF(u){\eta }_C={\eta }_{C^{\prime }}u$ for $u:C\to C^{\prime }$ and $v{\epsilon }_D={\epsilon }_{D^{\prime }}FG(v)$ for $v:D\to D^{\prime }$, so these are natural transformations $1_{\mathcal{C}}\to GF$ and $FG\to 1_{\mathcal{D}}$. More explicitly, naturality at $f:FC\to D$ and $g:C\to GD$ gives the two transpose formulasThese formulas will also prove the triangle identities for an adjunction.

Using the transpose formulas of the preceding part,Similarly,Thus the adjunction unit and adjunction counit satisfy both triangle identities for an adjunction,

Given the two natural transformations, defineNaturality of $\eta$ and the second triangle identity for an adjunction giveNaturality of $\epsilon$ and the first triangle identity for an adjunction giveThe maps are natural in $C,D$, since the defining compositions commute with precomposition in $C$ and postcomposition in $D$ by naturality of $\eta ,\epsilon$. They therefore define an adjunction $F⊣G$. Their values on identity morphisms recover precisely $\eta ,\epsilon$. Conversely any adjunction with these unit and counit of an adjunction must have the same transpose formulas, so it is unique with the prescribed unit and counit.