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