Let us fix two categories $\mathcal{C}$ and $\mathcal{D}$, as well as two functors $F:\mathcal{C}\rightarrow \mathcal{D}$ and $G:\mathcal{D}\rightarrow \mathcal{C}$. By an adjunction between $F$ and $G$ I mean a natural isomorphism of bifunctors $\Phi _{X,Y}:\mathrm{Mor}_{\mathcal{D}}(F(X),Y)\rightarrow \mathrm{Mor}_{\mathcal{C}}(X,G(Y))$. If there is some adjunction between $F$ and $G$, we say that $(F,G)$ is an adjointable pair.
Are there examples of adjointable pairs $(F,G)$, but for which the adjunction between them is not unique? If so, is there a sense in which they are all unique (analogous to, for example, how if a functor admits a left-adjoint, that functor is not literally unique, but it is unique up to natural isomorphism)?
My first thought is "Yes, there do exist such pairs.", and that you can find other adjunctions of the same adjointable pair by finding automorphisms of the identity functors on $\mathcal{C}$ and $\mathcal{D}$ (though I have not checked this in detail). If this is so, are adjunctions unique up to automorphisms of the identity functors, or we can find more adjunctions still?
Best Answer
Adjoint functors are not only unique up to isomorphism, they are unique up to unique isomorphism; that is, if $F$ is left adjoint to $G$ and also to $G'$, then there is a unique isomorphism $G \cong G'$ compatible with the data of the two adjunctions. In particular, this is true if $G = G'$. So it's indeed the case that adjunction data is unique up to twisting by an automorphism of $G$, or dually up to twisting by an automorphism of $F$ (in particular, $F$ and $G$ have the same automorphism group, but there's no reason that this should be trivial).