Characterize pullback functors among copresheaf-pra's

Question

Let $P$ and $Q$ be categories, and suppose $P$ has a terminal object $\ast_p$. A parametric right adjoint, or pra, $F$ from $P$ to $Q$ is a functor $F:P\to Q$ such that the functor $$F\,/\,{\ast_p}: P\,/\,{\ast_p}\to Q\,/\,F(\ast_p)$$ induced by slicing over $*_p$ is a right adjoint. In particular, since right adjoints preserve terminal objects, note that every right adjoint $P\to Q$ is a pra.

Suppose that $C$ and $D$ are categories, and let $P:=C{-}\mathsf{Set}$ and $Q:=D{-}\mathsf{Set}$ be the associated copresheaf categories. A functor $f\colon C\to D$ induces a pullback functor $$\Delta_f\colon D{-}\mathsf{Set}\to C{-}\mathsf{Set},$$ which has both a left adjoint $\Sigma_f$ and right adjoint

$$ \Pi_f\colon C{-}\mathsf{Set}\longrightarrow D{-}\mathsf{Set}. $$

In particular, each of $\Delta_f:Q\to P$ and $\Pi_f:P\to Q$ is a parametric right adjoint.

Remark: A functor $C{-}\mathsf{Set}\longrightarrow D{-}\mathsf{Set}$ is a pra iff it is isomorphic to one of the form $\Delta_e\,\overset{\circ}{,}\,\Pi_f\,\overset{\circ}{,}\,\Sigma_g$, where $$ C\xleftarrow{e}\bullet\xrightarrow{f}\bullet\xrightarrow{g}D $$ are categories and functors, where $(e,f)$ forms a two-sided discrete fibration, and where $g$ is a discrete opfibration. This fact is due to Mark Weber.

Definition: Let $\mathsf{pra}$ denote the category whose objects are categories $\text{Ob}(\mathsf{pra})=\text{Ob}(\mathsf{Cat})$ and for which a morphism $C\to D$ is a pra between their copresheaf categories $$ \mathsf{pra}(C,D):=\{F\colon C{-}\mathsf{Set}\longrightarrow D{-}\mathsf{Set}\;\mid\;F\text{ is a pra}\}. $$ We refer to $\mathsf{pra}$ as the category of categories and copresheaf-pras. It is in fact a bicategory whose 2-morphisms are natural transformations.

Question 1: Inside the bicategory $\mathsf{pra}$ of categories and copresheaf-pras, can one characterize those morphisms $F$ that are of the form $F=\Delta_f$ for some functor $f\colon D\to C$?

As pointed out by Simon Henry in an answer below, it is equivalent to ask:

Question 2: Inside the bicategory $\mathsf{pra}$ of categories and copresheaf-pras, can one characterize those morphisms $F$ that are of the form $F=\Pi_f$ for some functor $f\colon C\to D$?

Answer 1

A first remark is that question 1 and 2 are equivalent as in the category pra you have $\Delta_f \dashv \Pi_f$. So if you have a characterization of one class you characterize the other as their left/right adjoint functors

Now, the copresheaf functor from pra to Cat can be recovered as the global section functor:

Indeed, the terminal object of pra is the small category $\emptyset$, indeed $\emptyset$-Set is the terminal category and the unique functor $C$-Set $\to 1$ is always a (parametric) right adjoint functor.

Moreover, any functor $\emptyset$-Set $ =1\to C$-Set is parametric right adjoint, so you get that pra$(\emptyset$-Set,$C$-Set$) \simeq C$-Set.

So, at least assuming Cauchy completeness you can characterize the $\Delta_f$ and $\Pi_f$ as the adjunction $h \dashv g$ such that the action of $h$ on global section has a further left adjoint.

I don't think you can give a purely categorical characterization of these functor without assuming Cauchy-completness as every small category is isomorphic to its Cauchy-completion in pra: a purely categorical construction cannot distinguish between a $\Delta_f$ functor and something that is a $\Delta_f$ for an $f$ between the Cauchy completion.