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

An adjunction $F⊣G$ is a family of bijectionsnatural in both variables. Its adjunction unit and adjunction counit are ${\eta }_A={\Phi }_{A,FA}(1_{FA})$ and ${\epsilon }_B={\Phi }_{GB,B}^{-1}(1_{GB})$. Naturality of the correspondence givesFor $f:A\to A^{\prime }$, naturality in both variables evaluates $\Phi (Ff)$ in two ways, giving $GFf{\eta }_A={\eta }_{A^{\prime }}f$. Thus $\eta$ is a natural transformation; the dual calculation gives naturality of $\epsilon$.

Applying the inverse correspondence to ${\eta }_A$ and the correspondence to ${\epsilon }_B$ yields the triangle identities for an adjunction:They express that transposing an identity morphism and transposing back returns that identity.

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.

The currying adjunction for small categories uses the bijectionIt sends $H$ to the functor $d\mapsto H(d,-)$, with an arrow $u:d\to d^{\prime }$ inducing the natural transformation whose $c$-component is $H(u,1_c)$. Conversely, for $K:\mathcal{D}\to [\mathcal{C},\mathcal{E}]$, define its uncurried functor by $(d,c)\mapsto K(d)(c)$ andNaturality of $K(u)$ allows the two factors to be interchanged in the appropriate order, giving functoriality. The constructions are inverse and natural in $\mathcal{D}$ and $\mathcal{E}$. Hence $-\times \mathcal{C}$ is a left adjoint to $[\mathcal{C},-]$ on the category of small categories.

The unit sends $d$ to the functor $c\mapsto (d,c)$ and $u$ to the transformation $(u,1_c)$. The counit is evaluation $[\mathcal{C},\mathcal{E}]\times \mathcal{C}\to \mathcal{E}$: $(H,c)\mapsto H(c)$ and $(\alpha ,v)\mapsto H^{\prime }(v){\alpha }_c={\alpha }_{c^{\prime }}H(v)$.