Let $F : G\to PA$ be a functor ($G$ a small category, $PA = [A°,Set]$ the presheaf category of $A$). Let $N_F$ be the functor $PA\to PG$ defined by left Kan-extending the Yoneda embedding $y_G : G\to PG$ along $F$:
The following procedure seems to exhibit a right adjoint for $N_F$; this is strange, as $N_F$ is in general only a *right* adjoint. Can you spot the error (if any) in my argument?

Consider the extended diagram where the functor $PG \to PA$ is the left extension of $y_A$ along $N_F y_A$, that can easily shown to be equal to the left extension of the identity along $N_F$. And this latter functor shall be the

rightadjoint to $N_F$: counit and unit are determined by the suitable universal properties.

Many things can be true at this point:

- The functor does not exist.
- The functor exists, but fails to be a right adjoint to $N_F$ (the extension is not absolute/not preserved by $N_F$).
- The functor is indeed a right adjoint to $N_F$; I didn't know it had one in the special case where its domain is itself a free cocompletion.

Note that taking the *right* extension of $y_A$ along $N_Fy_A$ does not give a left adjoint to $N_F$: if such a left adjoint exists, it is unique and must exhbit the universal property of the left extension of $F$ along $y_A$:
$$
\text{Lan}_{y_A}F \dashv \text{Lan}_Fy_A
$$
(that's an instance of the $F$-nerve and $F$-oidal realization yoga). Indeed, a key step in the above argument is that $\text{Lan}_{N_F y_A}y_A\cong \text{Lan}_{N_F}(\text{Lan}_{y_A}y_A) \cong \text{Lan}_{N_F}1$, as $y_A$ is dense. But it's not codense: in fact, $\varphi \to \text{Ran}_{y_A}(y_A)(\varphi)$ is the $\varphi$-component of the unit of Isbell adjunction.