Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 1 / iii / Solution

Define maps between the hom-sets byUsing naturality of the proposed counit of an adjunction and then the first triangle identity for an adjunction,Using naturality of the proposed unit of an adjunction and then the other triangle identity for an adjunction,Hence $\Phi$ and $\Psi$ are inverse bijections. They are natural in both variables: for $a:A^{\prime }\to A$ and $b:B\to B^{\prime }$, the unit of an adjunction identity givesThis constructs an adjunction $F⊣G$. Taking $f=1_{FA}$ and $g=1_{GB}$ recovers the prescribed unit of an adjunction and counit of an adjunction. Conversely, part (i) shows that any adjunction with those components must have exactly these transpose formulas. The adjunction exists and is unique with the prescribed unit and counit.