Are left and right adjoint functors of a functor natural equivalent?

Question

Q: If a functor has both left and right adjoint functors, then whether there exists a natural equivalence between the two adjoint functors?

Any help would be appreciated.

Answer 1

No, the left and right adjoints to a functor (if they exists) are usually not isomorphic.

For example, if $\mathcal{C}$ is a category with finite limits and colimits and $\mathcal{K}$ is a finite category, then the diagonal functor $\Delta\colon \mathcal{C}\to \mathcal{C}^{\mathcal{K}}$, which maps an object $C$ to the constant functor $\Delta_C$ defined by $\Delta_C(K)=C$ for all objects $K$ in $\mathcal{K}$ has a left and a right adjoint, which are the colimit and limit functors $\mathcal{C}^{\mathcal{K}}\to \mathcal{C}$. Since the colimit of a functor is usually different from its limit, these adjoints are not isomorphic.