Let $C$ and $D$ be categories and $F:C\to D$, $G:D\to C$ two functors.
$F$ is left-adjoint to $G$, if there are natural transformations $\eta:id_C\to GF$ and $\epsilon:FG\to id_D$ such that
\begin{eqnarray}
F&\xrightarrow{F\eta}&FGF&\xrightarrow{\epsilon F}F\\
G&\xrightarrow{\eta G}&GFG&\xrightarrow{G\epsilon}G
\end{eqnarray}
are the identity (!) transformations.
$F$ is an equivalence of categories (with inverse $G$) if there are natural isomorphisms $\eta:id_C\to GF$ and $\epsilon:FG\to id_D$ without any further properties.
Is $F$ left-adjoint to $G$, if $F$ is an equivalence of categories (with inverse $G$)?
If not, suppose that $F$ is an equivalence of categories with inverse $G':D\to C$ and suppose further that $F$ is left-adjoint to $G$. Does it follow that there is a natural isomorphism $G\to G'$ or is there even an identity $G=G'$?
Best Answer
Every equivalence can be "improved" to an adjoint equivalence, but you may have to change the unit or the counit: see e.g. here for a proof. Essentially the same thing is proved as Theorem 4.2.3 in the Homotopy Type Theory book.