TITLE: A specific property of bi-adjunction
QUESTION [5 upvotes]: Let $$I: C \rightleftarrows D: F$$ be biadjoint [1] functors between categories $C, D$. That is, $I$ is the left and also the right adjoint of $F$ (thus vice versa). Put in notations, it's
$$ \cdots \vdash I \vdash F \vdash I \vdash F \cdots$$
Then since $I$ is left adjoint to $F$, for each object $X$ in $D$ there exists a unit map
$$ \eta_X: X \to IFX.$$
Similarly, since $F$ is left adjoint to $I$, there exists a unit map
$$ \eta'_{FX}: FX \to FIFX.$$
Question: Given their type, one naturally asks if $F(\eta_X): FX \to FIFX$ equals to $\eta'_{FX}$, or is there any general relation?
(Or if you have some interesting examples of such functors, please let me know. I'm happy to check if they satisfy this condition or not.)
Attempt: This is a special case of an ambidextrous adjunction. The corresponding (co)monads are Frobenius monads. I've looked in some paper about them, and couldn't find this structure discussed.
I've also tried to prove it myself.. but these unit maps come from different adjunctions, so a priori there is no reason for them to relate. But I can't be sure.
Footnote
[1] "biadjunction" is not a standard term. see Mike Shulman's comment below

REPLY [4 votes]: Qiaochu Yuan’s answer excellently explains the general phenomenon.  Here is another slightly simpler concrete example:
Let $\newcommand{\Z}{\mathbb{Z}}\newcommand{\BZ}{{\mathbf{B}\Z}}\BZ$ be the group $\Z$ viewed as a one-object groupoid, with arrows $a^n : \ast \to \ast$ for $n \in \Z$.  Then its identity functor $\newcommand{\id}{\mathrm{id}}\id_\BZ$ is self-adjoint in $\Z$-many ways: for any $n$, we can take $a^n$ as the unit and $a^{-n}$ as the counit.  (Any identity functor is trivially self-adjoint, and then as Qiaochu explains, we can conjugate it by natural automorphisms of the identity functor, which in this case are precisely $\Z$.)
So now we can choose units+counits for the biadjunction $\id_\BZ \dashv \id_\BZ \dashv \id_\BZ$ in $\Z^2$-many ways.  Writing $a^n$ and $a^m$ for the two choices of unit, your condition will hold just when $m=n$.