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It's trivially possible for a functor to be both a left and right adjoint functor paired with different functors. But can they be to the same category? In other words, can one find a $G:C\to D$ which is a right adjoint to $F:D\to C$ and a left adjoint to $H:D\to C$ other than the trivial case where $G$ is a bijection and $F$ and $H$ are its inverse?

My instinct is that a category cannot be used both ways, but I'm having trouble proving it.

TShiong
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Cort Ammon
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There is plenty of examples. Take for instance the subcategory of groupoids in small categories $\mathrm{Grpd} \hookrightarrow \mathrm{Cat}$. This inclusion is bireflective, i.e. it has both left and right adjoints. The left adjoint is the full localisation of a subcategory $\mathcal C[\mathcal C^{-1}]$, which has the same objects with additional free inverse of all the arrows. On the other hand its right adjoint is the core groupoid, again with the same objects, but non-invertible arrows removed

Alexander Golys
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  • Is "bireflective" the keyword I should have been searching for, based on the description I gave? – Cort Ammon Apr 08 '23 at 23:41
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    A subcategory is called reflective is the inclusion has left adjoint and coreflective if it has right adjoint. Subcategories both reflective and coreflective are sometimes called bireflective – Alexander Golys Apr 09 '23 at 04:01