It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "[nerve-realization](https://ncatlab.org/nlab/show/nerve+and+realization)" adjunction
$$
\text{Lan}\_y F \dashv N\_F = \hom(F,1)
$$
that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as
$$
\text{Lan}\_y F \dashv \text{Lan}\_F y,
$$
and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[[here](https://golem.ph.utexas.edu/category/2014/03/an_exegesis_of_yoneda_structur.html), several comments in the discussion are mine]. 


1. Is there a reason why this is true?
2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}\_GF\dashv \text{Lan}\_FG$?
3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N\_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N\_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints
$$
\text{Lan}\_yF = \varphi\_F^! \dashv \varphi^F\dashv \varphi\_F.
$$
What's the meaning of this extension, and its universal property, in $\bf Prof$?



The Kan construction, profunctors, and Kan extensions
I will try to answer the second question. 



> 
> **Prop 1.** Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}\_if$ is pointwise. Then, the following are equivalent.
> 
> 
> 1. $\text{lan}\_if \dashv \text{lan}\_fi$.
> 2. $f$ is the $i$-relative left adjoint of $\text{lan}\_fi$, i.e. ${\bf C}(i, \text{lan}\_fi) \cong {\bf B}(f, - ).$
> 3. $f = \text{lift}\_{\text{lan}\_fi}i$ and the lift is absolute.
> 
> 
> If $i$ is only fully faithful $1 \Rightarrow 2$, if $i$ is only dense $2 \Rightarrow 1$.
> 
> 
> 


Proof.


$1 \Rightarrow 2$) $${\bf B}(f, -) \stackrel{i \text{ is ff.}}{\cong} {\bf B}((\text{lan}\_if) i, -) \stackrel{1}{\cong} {\bf C}(i, \text{lan}\_fi).$$


$2 \Rightarrow 1$)
$${\bf B}(\text{lan}\_if, -) \stackrel{\text{point.}}{\cong} \text{lan}\_i{\bf B}(f, -) \stackrel{2}{\cong} \text{lan}\_i{\bf C}(i, \text{lan}\_fi) \stackrel{\text{point.}}{\cong} {\bf C}(\text{lan}\_ii, \text{lan}\_fi) \stackrel{i \text{ is dense}}{\cong} {\bf C}(-, \text{lan}\_fi).$$


$3$ is just a rewriting of $2$.




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Now we study a very special setting. 


Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}\_if$ is pointwise, ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete.


In this setting ${\bf C}$ is a reflective subcategory $ V: {\bf C} \leftrightarrows \text{Set}^{{\bf A}^\circ} : L $ of $\text{Set}^{{\bf A}^\circ}$ via the nerve $V = \text{lan}\_i(y\_{{\bf A}})$ (V is the right adjoint).



> 
> **Prop 2.** Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete. Then, the following are equivalent.
> 
> 
> 1. $\text{lan}\_if \dashv \text{lan}\_fi$.
> 2. V preserves $\text{lan}\_fi$.
> 3. ${\bf C}(i,\text{lan}\_fi) \cong \text{lan}\_f{\bf C}(i,i)$.
> 
> 
> 


Proof.
Recall that $\text{lan}\_if$ is pointwise,because ${\bf B}$ is cocomplete.


$1 \Rightarrow 2)$.


Using Prop 1. we know that ${\bf C}(i, \text{lan}\_fi) \cong {\bf B}(f, - )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y\_A, \text{lan}\_fy\_A) \cong {\bf B}(f, -)$.


Thus, $$ {\bf C}(i, \text{lan}\_fi) \cong {\bf B}(f, -) \cong \text{Set}^{{\bf A}^\circ}(y\_A, \text{lan}\_fy\_A) \cong \text{Set}^{{\bf A}^\circ}(y\_A, \text{lan}\_fVi)$$


Observe also that $${\bf C}(i, \text{lan}\_fi) \cong {\bf C}(Ly\_A, \text{lan}\_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y\_A, V\text{lan}\_fi),$$ putting the last two equation together, one gets: $$\text{Set}^{{\bf A}^\circ}(y\_A, \text{lan}\_fVi) \cong \text{Set}^{{\bf A}^\circ}(y\_A, V\text{lan}\_fi). $$ By Yoneda Lemma the two functors on the right have to coincide.


$2 \Rightarrow 1).$


Using Prop 1. it is enough to prove that ${\bf C}(i, \text{lan}\_fi) \cong {\bf B}(f, - )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y\_A, \text{lan}\_fy\_A) \cong {\bf B}(f, -)$. Now, $${\bf C}(i, \text{lan}\_fi) \cong {\bf C}(Ly\_A, \text{lan}\_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y\_A, V\text{lan}\_fi) \stackrel{2}{\cong} \text{Set}^{{\bf A}^\circ}(y\_A, \text{lan}\_fVi) \cong \text{Set}^{{\bf A}^\circ}(y\_A, \text{lan}\_fy\_A) \cong {\bf B}(f, -). $$


$3$ is just a rewriting of $2$.




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> 
> **Cor. 1** (Quite surprising).
>  Let ${\bf Set}^{A^\circ} \xleftarrow{y} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $y$ is the Yoneda embedding of a small category and ${\bf B}$ is cocomplete. Then $\text{lan}\_yf \dashv \text{lan}\_fy$.
> 
> 
> 


Proof 1. In Prop 1 the condition 2 is verified and is essentially a rewriting of the Yoneda Lemma. Moreover $\text{lan}\_yf$ is pointwise because ${\bf B}$ is cocomplete.


Proof 2. In Prop 2 the condition 2 is trivially verified because $V$ is the identity.




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I shall conclude by saying that the adjunction cannot verify too often.



> 
> **Rem. 4** Let $y: {\bf A} \to {\bf C}$ be a dense and fully faithful functor. Then the following are equivalent.
> 
> 
> 1. For every map $f: {\bf A} \to {\bf B}$, where ${\bf B}$ is cocomplete, the Kan extensions $\text{lan}\_yf$ and $\text{lan}\_fy$ exist and are adjoint.
> 2. $y$ is the Yoneda embedding.
> 
> 
> 


Proof. One implications is Cor 1. The other implication can be read as follows, if 1 is verified, then for every functor $f: {\bf A} \to {\bf B}$, there is a unique cocontinous extension ($\text{lan}\_yf$), this is the characterization of the free cocompletion under colimits, that is the (small) presheaf construction.


