Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 18 / 5 / i / Solution

Write the adjunction bijections as ${\Phi }_{A,B}:\mathcal{D}(FA,B)\cong \mathcal{C}(A,GB)$, natural in $A,B$. The adjunction unit and adjunction counit areThus $\eta :1_{\mathcal{C}}\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1_{\mathcal{D}}$ are natural transformations, by naturality of the bijections. For any $h:FA\to B$ and $k:A\to GB$, their transposes satisfyThe first follows by naturality in the target applied to $1_{FA}$; the second follows by naturality in the source applied to $1_{GB}$. These formulas make the definitions operational for every arrow.