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

Write the adjunction as the natural bijectionThe unit of an adjunction and counit of an adjunction are the natural transformations whose components areNaturality of the hom-set bijections gives, for $f:FA\to B$ and $g:A\to GB$,For example, postcomposition by $f$ in the first hom-set corresponds to postcomposition by $Gf$ in the second, giving the first formula; naturality in $A$ gives the second. The same naturality says that, for $a:A\to A^{\prime }$ and $b:B\to B^{\prime }$,Thus these components do define the asserted natural transformations $1_{\mathcal{C}}\to GF$ and $FG\to 1_{\mathcal{D}}$.

The two formulas for the adjunction transposes give both triangle identities for an adjunction directly. Apply the inverse bijection to the transpose of $1_{FA}$:Apply the forward bijection to the inverse transpose of $1_{GB}$:These are equalities of components of natural transformations, soNo cancellation assumption on either functor is needed: both identities follow from the inverse hom-set bijections.

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.