Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 18 / 4 / a

Write $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$. The characterization concerns an adjunction with the specified unit and counit of an adjunction $\eta$ and $\epsilon$. Its triangle identities for an adjunction areAssume these identities. Define the hom-set mapsNaturality of $\eta$ and $\epsilon$ makes these maps natural in both objects. Naturality and the triangle identities for an adjunction giveThus they are inverse bijections and define $F⊣G$.

Conversely, from the natural hom-set bijections of an adjunction, define ${\eta }_A=\Phi (1_{FA})$ and ${\epsilon }_D={\Phi }^{-1}(1_{GD})$. Naturality gives the same formulas for $\Phi$ and $\Psi$ above. Applying $\Psi \Phi$ to $1_{FA}$ and $\Phi \Psi$ to $1_{GD}$ gives the two triangle identities for an adjunction. Therefore these identities are exactly the compatibility conditions on the specified unit and counit. If the printed equivalence were read as mere existence of some adjunction, independently of the supplied transformations, its only-if direction would be false: on the category of abelian groups, $F=G=1$ are adjoint, but choosing both transformations to be zero does not satisfy either triangle on a nonzero object.