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Is there a good example of an equivalence of categories $F:\mathcal{A}\to\mathcal{B}$ such that $F$ is neither left nor right adjoint to its inverse $G:\mathcal{B}\to\mathcal{A}$?

mez
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    Every such equivalence can be improved to an adjoint equivalence where $F$ and $G$ do form an adjoint pair. – Aleš Bizjak Jan 07 '15 at 13:28
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    Every equivalence $(F,G)$ of categories comes with natural isomorphisms $\eta:\Bbb 1_A\Rightarrow GF$ and $\eta:FG\Rightarrow\Bbb 1_B$. One can choose this $\eta$ to be the unit of the adjunction where $F$ is left adjoint to $G$. As a matter of fact, $\eta^{-1}$ is then the counit of the adjunction where $G$ is the left adjoint of $F$. The functors and the choice of unit then determine the counit uniquely. – Stefan Hamcke Jan 08 '15 at 17:52

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