A functor that has both left and right adjoints

Question

What can we say about a functor that has both left and right adjoints?

I vaguely recall hearing that it is then an equivalence of category. Is it true?
If not, then under what conditions it is true?

Answer 1

No, it's not necessarily an equivalence of categories. For example, if $\varphi: R\to S$ is a homomorphism of rings, the forgetful functor $S\text{-Mod}\to R\text{-Mod}$ always has both the left adjoint $\text{Ind}_R^ S := -\otimes_R S$ and the right adjoint $\text{CoInd}_R^S:=\text{Hom}_R(S,-)$. It can even happen that these coincide without $S\text{-Mod}\to R\text{-Mod}$ being an equivalence: for example, if $R={\mathbb k}$ is a field and $S = {\mathbb k}G$ is the group algebra of a finite group $G$ over ${\mathbb k}$, then $\text{Ind}_{\mathbb k}^{{\mathbb k}G}\cong\text{CoInd}_{\mathbb k}^{{\mathbb k}G}$, but still ${\mathbb k}G\text{-Mod}\to{\mathbb k}\text{-Mod}$ is not an equivalence if $|G|>1$.

It also happens often that the inclusion of a subcategory has both a left and a right adjoint.

I don't know of a useful criterion telling you that a functor which has both left and right adjoints + satisfies certain extra properties has to be an equivalence of categories.