Does the nerve have a right adjoint, sometimes?

Question

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

Many things can be true at this point:

1. The functor does not exist.
2. The functor exists, but fails to be a right adjoint to $N_F$ (the extension is not absolute/not preserved by $N_F$).
3. 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.

Answer 1

To answer the title question, but not the question about your argument....

$PG$ and $PA$ are locally presentable categories, so $N_F : PA \to PG$ has a right adjoint if and only if it preserves small colimits.

From the formula $$ N_F(Q)(g) \cong \hom_{PA}(F(g), Q) $$ the condition that $N_F$ preserves colimits is equivalent to natural isomorphisms $$ \hom_{PA}(F(g), \operatorname{colim}_j Q_j) \cong \operatorname{colim}_j N_F(Q_j)(g) \cong \operatorname{colim}_j \hom_{PA}(F(g), Q_j) $$

In other words, $F(g)$ must be a tiny object for every $g$.

But in a presheaf category, the tiny objects are precisely the retracts of representables. If the domain category is idempotent complete, this simplifies to the tiny objects being precisely the representables.

In other words, $N_F$ has a right adjoint if and only if $F$ can be written as a composite of a functor $G \to \operatorname{Idem}(A)$ followed by the yoneda embedding.

Answer 2

In light of user13113's observation that a nerve $N_F$ is right adjoint only if $F$ is the composite of some functor $L$ followed by the Yoneda embedding $y_A$, your observation is essentially contained in Lemma 13 of Walker's Yoneda structures and KZ doctrines. In Lemma 13, Walker proves that $N_F$ (for $F \cong y_A L$) admits a right adjoint when $N_L$ admits a nerve, in which case $N_F \dashv N_{N_L}$, which recovers precisely your expression for the right adjoint.