Does there exist a category $\mathcal{C}$ and contravariant functors $F:\mathcal{C}\to\mathcal{C}^{op}, G:\mathcal{C}^{op}\to\mathcal{C}$ such that $F$ is left ajoint to $G$ and $F\neq G^{op}$? In the case of $\mathcal{C}=Set$, I proved that there exist no such functors, because if exist $G$ preserve limits, and so is representable by adjoint functor theorem, and that $G=Hom(-,A)$ has a left ajoint $F=Hom(-,A)$ itself. But I can't come up with any answers to other cases. Partial solutions are welcomed. For example, when $\mathcal{C}$ is a topos, or $G$ is monadic, etc.



Adjoint functors between dual categories
Here's a very simple example. Let $\mathcal{C}$ be any set with more than two elements, considered as a discrete category. Then $\mathcal{C}=\mathcal{C}^{op}$, and any permutation $F:\mathcal{C}\to\mathcal{C}^{op}=\mathcal{C}$ has an adjoint $G$ (on both sides!) given by the inverse. Any permutation that is not its own inverse then gives an example of an adjunction with $F\neq G^{op}$.


