Description of adjunctions in a 2-category via Kan extensions? Let $\Psi$ be a strict (for now) 2-category.

*

*Let $C,D$ be 0-cells in $\Psi$;

*let $L : C \rightarrow D$
and $R : D \rightarrow C$ be 1-cells;

*and let $\eta : 1_C \Rightarrow RL$ and $\epsilon : LR \Rightarrow 1_D$ be 2-cells.


Claim:
Then, if I'm not mistaken, the following claim holds:
$$
\left(  L \dashv R \textrm{ in $\Psi$ via $\eta,\epsilon$} \right) 
\Leftrightarrow
\left( 
\begin{matrix}
R = \mathrm{Lan}_L(1_C) \textrm{ via $\eta$ in $\Psi$} \\
\textrm{&} \\
L = \mathrm{Ran}_R(1_D) \textrm{ via $\epsilon$ in $\Psi$}
\end{matrix}
\right) 
$$

Here:

*

*the LHS is using the definition of an adjunction of 1-cells in a 2-category (via unit and counit 2-cells), as in e.g. this article.

*the RHS is using the general notion of left/right Kan extensions of 1-cells in a 2-category, as discussed at e.g. this article.


Question 1: Is the above claim correct for a strict $2$-category? Why or why not?

If the answer to this question is "yes", then

Question 2: Does the above claim generalize to the case when $\Psi$ is a bicategory?


Note for Question 1:
I'm self-studying some (2-)category theory, and I think I've managed (following an answer from the post Adjunctions are Kan Extensions. for the $\Rightarrow$ direction) to prove the above, directly using the zig-zag identities defining adjunctions and the defining properties for Kan extensions.
Note for Question 2:
(Unless I'm mistaken, the notions of adjunction and Kan extension also work in a bicategory. (More specifically, I think for adjunctions we just need to introduce unitors in the zig-zag diagrams, and for Kan extensions the definition works verbatim in a bicategory?) Here for the definition of bicategory I'm following this article.)
 A: Update: so after some more reading, I think if the above claim does extend to the bicategory situation, then a possible tool to help prove it might be the coherence theorem for bicategories (generalizing MacLane's coherence theorem for monoidal categories)? As discussed at https://ncatlab.org/nlab/show/bicategory#Coherence.
However, I don't yet fully understand the coherence theorem nor precisely how to use it here, so I guess I'll consider the question yet unanswered.

Also I'm still not sure whether the result claimed above is true in a (strict) 2-category. For some reason I can't seem to find this specific result written down in any sources, even though there is plenty written about adjoints in relation to Kan extensions; although it's possible I just haven't been looking in the right places.
A: I think the $2$-category freely generated by two objects $C$ and $D$, two $1$-morphisms $L\colon C\to D$ and $R\colon D\to C$ and two $2$-morphisms $\eta\colon 1_C\to RL$ and $\epsilon\colon LR\to 1_D$ is a counter-example to the claim.
Indeed, $1$-moprhisms in this $2$-category are $1_C$, $1_D$, and finite alternating sequences $LRLRLR...$ or $RLRLRLRL..$. The $2$-morphisms are then sequences starting with $1_C$ or an alternating sequence, then successively either inserting an $RL$ into the alternating sequence, or removing an $LR$ from the alternating sequence, until ending at an alternating sequence or $1_D$. Two such sequences define equal $2$-morphisms if they are made equivalent by swapping successive insertions and removals that do not affect overlapping pairs of $RL$ and $LR$ in the sequence.
Evidently, any $2$-morphism $1_C\Rightarrow\dots L$ has to start with inserting an $RL$. Further, this $L$ stays as the right-most $L$ unless an $RL$ is inserted after it, in which case the $2$-morphism constructed so far is equal to the constructed by doing that insertion first. Thus, any $2$-morphism $1_C\Rightarrow RL$ proceeds with inserting the $RL$ in the right-most position, hence factors as $1_C\Rightarrow RL\to\dots L$ for a unique $2$-morphism $R\to\dots$.
Thus the unit is a unit of a left Kan extension of $1_C$ along $L$. Similarly, the counit is that of a right Kan extension of $1_D$ along $R$.
But the unit and counit evidently do not satisfy the zig-zag/triangle identities.

What actually is going wrong is that adjoint functors are not simply Kan extensions but absolute Kan extensions (i.e. preserved by post-composition with every $1$-morphism). Adjoints are also absolute Kan lifts and the example above shows that absoluteness is not guaranteed even if we assume the unit and counit are give us both Kan extensions and Kan lifts.
In fact, absoluteness amounts to the zig-zag/triangle identies as follows (in the $2$-categorical setting, for bicategories perhaps inserting the appropriate unitors and associators below works, but I haven't worked that out).
Proposition. For $1$-morphisms $L\colon C\to D$, $R\colon D\to C$, and a $2$-morphism $\eta\colon 1_C\Rightarrow RL$, the following are equivalent:

*

*For every $1$-morphism $J\colon B\to C$ and $1$-morphism $H\colon B\to D$, any $2$-morphism $u\colon J\Rightarrow RH$ factors as $Ru'\circ \eta J\colon J\Rightarrow RLJ$ for some $2$-morphism $u'\colon LJ\Rightarrow H$.


*There exists a $2$-morphism $\epsilon\colon LR\Rightarrow 1_D$ so that $\mathrm{id}_R\colon R\Rightarrow R$ factors as $R\epsilon\circ\eta R\colon R\Rightarrow LRL\Rightarrow R$. Moreover, in this case



*

*For every $1$-morphism $K\colon C\to E$, and for every pair of $2$-morphisms $w_1,w_2\colon KR\Rightarrow H\colon D\to E$, we have that $w_1L\circ K\eta=w_2L\circ K\eta$ implies $w_1=w_2$. In other words, $K\eta\colon K\Rightarrow KRL$ is a unit that makes $KR\colon D\to E$ satisfy the uniqueness property of a left Kan extension of $K\colon C\to E$ along $L\colon C\to D$.

*$\mathrm{id}_L\colon L\Rightarrow L$ factors as $\epsilon L\circ L\eta=\mathrm{id}_L\colon L\Rightarrow LRL\Rightarrow L$ if the factorizations in 1. are unique when $J=R\colon D\to C$ and $H=1_D\colon D\to D$.

Proof. Clearly 1. implies 2 by taking $J=R$, $H=1_D$, and $u=\mathrm{id}_R\colon R\Rightarrow R$, in which case $\epsilon=u'$. Conversely, 2. implies
$u=\mathrm{id}_RH\circ u=R\epsilon H\circ\eta RH\circ u=R\epsilon H\circ\eta u=R\epsilon H\circ RLu\circ\eta J$, so $u'=R(\epsilon H\circ Lu)$ gives 1. Moreover, 2 implies that for $w_i\colon KR\to H$ we have $w_i=w_i\circ K\mathrm{id}_R=w_i\circ KR\epsilon\circ K\eta R=H\epsilon \circ w_iLR\circ K\eta R=H\epsilon\circ(w_iL\circ K\eta)R$ whence that $w_1L\circ K\eta=w_2L\circ K\eta$ implies $w_1=w_2$. Finally, 2. imples that $R(\epsilon L\circ L\eta)\circ\eta R=R\epsilon L\circ RL\eta \circ\eta=R\epsilon L\circ\eta RL\circ\eta=(R\epsilon\circ\eta R)L\circ\eta=\mathrm{id}_R\circ\eta R$, so uniqueness in 1. would imply $\epsilon L\circ L\eta=\mathrm{id}_L$.
Note that we can rephrase 1. as the assertion that $\eta\colon 1_C\Rightarrow RL$ is the unit of an absolute weak left Kan lift $L\colon C\to D$ of $1_C\colon C\to C$ along $R\colon D\to C$. Here "weak" refers to only requiring existence rather than uniqueness of factorizations, and "absolute" refers to being stable under pre-composition.
Similarly, let's say that $\epsilon\colon LR\Rightarrow 1_D$ is the counit of an absolute wak right Kan extension $L\colon C\to D$ of $1_D\colon D\to D$ along $R\colon D\to C$ if for every $1$-morphism $K\colon D\to E$ and $1$-morphisms $H\colon C\to E$, any $2$-moprhism $e\colon HR\Rightarrow K$ factors as $K\epsilon\circ e'R\colon HR\Rightarrow KLR\Rightarrow K$ for a unique $2$-morphism $e'\colon H\Rightarrow KL$.
Applying co-op-duality (formally flipping the directions of the $1$-moprhisms and of the $2$-morphisms), we obtain
Proposition.
For $1$-morphisms $L\colon C\to D$, $R\colon D\to C$, and a $2$-morphism $\epsilon\colon LR\Rightarrow 1_D$, the following are equivalent:

*

*$\epsilon\colon LR\Rightarrow 1_D$ is the counit of an absolute weak right Kan extension $L\colon C\to D$ of $1_D\colon D\to D$ along $R\colon D\to C$.

*There exists a $2$-morphism $\eta\colon 1_C\Rightarrow RL$ so that $\mathrm{id}_R\colon R\Rightarrow R$ factors as $R\epsilon\circ\eta R\colon R\Rightarrow LRL\Rightarrow R$. Moreover, in this case


*

*For every $1$-morphism $J\colon B\to D$ and for every pair of $2$-morphisms $d_1,d_2\colon H\Rightarrow RJ$, we have that $Ld_1\circ\epsilon J=Ld_2\circ\epsilon J$ implies $d_1=d_2$. In other words, $\epsilon J\colon LRJ\to J$ is a counit that makes $RJ\colon B\to C$ satisfy the uniqueness property of a right Kan lift of $J\colon B\to C$ along $L\colon C\to D$.

*$\mathrm{id}_L\colon L\Rightarrow L$ factors as $\epsilon L\circ L\eta\colon L\Rightarrow LRL\Rightarrow L$ if $R\epsilon\colon RLR\Rightarrow R$ is the counit of a right Kan extension of $R\colon D\to C$ along itself.

Combining the two propositions, we see that the zig-zag/triangle identity $R\epsilon\circ\eta R=\mathrm{id}_R$ has the following consequences:

*

*$\eta\colon 1_C\Rightarrow RL$ is the unit of an absolute weak left Kan lift $L$ of $1_C$ along $R$

*$\eta\colon 1_C\Rightarrow RL$ is the unit of an absolute "extremal" left Kan extension $R$ of $1_D$ along $L$ ("extremal" meaning uniqueness but not existence)

*$\epsilon\colon LR\Rightarrow 1_D$ is the counit of an absolute weak right Kan extension $L$ of $1_D$ along $R$

*$\epsilon\colon LR\Rightarrow 1_D$ is the counit of an absolute "extremal" right Kan lift $R$ of $1_C$ along $L$
Another application of duality shows that the other zig-zag/triangle identity $\epsilon L\circ L\eta=\mathrm{id}_L$ has the same consequences with "weak" and "extremal" switched ("extremal" is non-standard term, but I haven't found a better one). Since being "weak" and "extremal" means existence and uniqueness of factorization property, we get the equivlence of adjoint functors with being absolute Kan extensions or absolute Kan lifts.
Moreover, when one zig-zag/triangle identity holds, the other holds if and only if one of its corresponding absolute weak Kan lift or extension is an absolute Kan lift or extension.
