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I am quite confused over what it means for a functor to have "both adjoints". From what I understand. Suppose we have $F:X \rightarrow A$ and $G:A\rightarrow X$. We might say $F$ is the left adjoint to $G$, thus $G$ have a left adjoint. Or we can say $G$ is the right adjoint to $F$, thus $F$ has a right adjoint.

My confusion is to what we mean, say, for $F:X\rightarrow A$ to have "both" adjoints ?

Cheers !

user16319
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    It means that $F$ has a left adjoint $G$ and also a right adjoint $H$. – Randall Nov 21 '19 at 14:15
  • @Randall. Thank you for the reply, my source of the confusion is the statement "$F$ has a left adjoint G", it looks like $G$ is also a functor from $X \rightarrow A$. – user16319 Nov 21 '19 at 14:20
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    No, if $F: X \to A$ and $G$ is left adjoint to $F$, then $G: A \to X$. If $H$ is right adjoint to $F$ then $H: A \to X$, too, but this need not be related to $G$. – Randall Nov 21 '19 at 14:25
  • See here for an example. – Arnaud D. Nov 21 '19 at 15:14

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