Can a functor $F$ between distinct categories be both left and right adjoint to the same functor G?

Question

I know that there are endofunctors that are self-adjoint. Here, I am not interested in this case.

So, suppose that $\mathcal{C}$ and $\mathcal{D}$ are "distinct" categories, and $F:\mathcal{C}\to\mathcal{D}$ a functor. Could there exist a functor $G:\mathcal{D}\to\mathcal{C}$, that is both a right- and left- adjoint of $F$?

I put scare quotes around "distinct" above because there's the possibility of isomorphism between $\mathcal{C}$ and $\mathcal{D}$. Since endofunctors can be self-adjoint, I suspect that the answer to this question will depend on whether the "distinct" qualification really means "non-isomorphic". Please consider this aspect in your answer.

NB: my question is different from this earlier question because in mine only two functors are involved.

Look up dagger-categories. – Martin Brandenburg Nov 14 '24 at 16:26 — Martin Brandenburg, Nov 14 '24 at 16:26

score 3 · Accepted Answer · answered Nov 14 '24 at 13:43

3

Yes: these are sometimes called Frobenius pairs.

answered Nov 14 '24 at 13:43

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Can a functor $F$ between distinct categories be both left and right adjoint to the same functor G?

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