Uniqueness of adjoints

Remarks

For each $d$, the adjunctions give bijections $\Hom(c, G d) \cong \Hom(F c, d) \cong \Hom(c, G' d)$, natural in $c$. By the Yoneda lemma, these come from a unique isomorphism $G d \cong G' d$. Naturality in $d$ makes these isomorphisms a natural isomorphism. The isomorphism is the unique one compatible with the two adjunctions.

Example: the free group on a set can be built from reduced words or as a quotient of a free monoid. Both constructions are left adjoint to the forgetful functor, so they give isomorphic functors. Likewise the abelianization $H \mapsto H/[H, H]$ is determined up to isomorphism by being left adjoint to the inclusion of abelian groups into groups.

Going deeper

For a student

The proposition is a form of the uniqueness of objects with a universal property, applied object by object. It lets one define a functor by asking that it be an adjoint, as for the tensor product, the abelianization or the Stone–Čech compactification. It also shows that adjunctions compose up to isomorphism: if $F \dashv G$ and $F' \dashv G'$, then $F' F \dashv G G'$, and any other left adjoint of $G G'$ is isomorphic to $F' F$.