Let $T\colon A\to B$ be a functor such that the category $A$ has a terminal object $1$. Then $T$ can canonically be factored as the composite
of $T$ applied to the slice category $A \simeq A/1$, followed by dependent sum (projection on the source).
We say that $T$ is a parametric right adjoint, or p.r.a., if the functor $T_1$ is a right adjoint. Parametric right adjoints are also called local right adjoints.
A monad is called p.r.a. if its functor part is p.r.a. and moreover its unit and multiplication are cartesian. Thus in particular it is a cartesian monad. A p.r.a. monad is also called a strongly cartesian monad.
Since $\Sigma_{T1}$ creates connected limits, if $T$ is p.r.a. then it preserves connected limits, and in particular preserves pullbacks. It follows that any p.r.a. monad is a cartesian monad.
Conversely, a functor between presheaf categories is p.r.a. if it preserves connected limits. The reason is that (in the notation above), if $T$ preserves connected limits, then $T_1$ preserves all small limits, and a limit-preserving functor from a cototal category such as a presheaf category to a locally small category is a right adjoint.
Any polynomial functor $\Sigma_h \Pi_g f^*$ is p.r.a., since then $T_1$ can be identified with $\Pi_g f^*$, which has the left adjoint $\Sigma_f \; g^*$.
If $E$ is a presheaf category and $T\colon E \to Set$ is p.r.a., then the comma category $Set/T$ (also called the Artin gluing in this context) is again a presheaf category. Conversely, if $Set/T$ is a presheaf category, then $T$ preserves connected limits, and thus is p.r.a.
A parametric right adjoint functor (with locally small codomain) has in particular a left multi-adjoint, which sends each object $b\in B$ to the family of all units $\eta_{b,i} : b \to T L(b,i)$, where $i$ ranges over all morphisms $b\to T 1$ and $L : B/T1 \to A$ is the left adjoint of $T_1$. This is because any morphism $b\to T a$ induces a unique composite $i:b \to T a \to T 1$, and hence a unique factorization through $L(b,i)$. Conversely, if $A$ has a terminal object and $T:A\to B$ has a left multi-adjoint, then it is a parametric right adjoint.
Central to the theory of parametric right adjoints is the notion of $T$-generic morphisms. For any functor $T$, a morphism $f \colon B\to T A$ is (strictly) $T$-generic if any commutative square of the following form:
has a unique filler of the form $T\delta \colon T A \longrightarrow T X$.
A generic factorization of a map $f\colon B\to T A$ is a factorization
such that $g$ is $T$-generic. Note that by the definition of genericity, generic factorizations are unique whenever they exist. If $T$ is a monad and any map $B \to T A$ has a generic factorization, then there is an induced orthogonal factorization system on the Kleisli category of $T$ in which $T$-generic maps are the left class and the right class are the “free” maps, i.e. those which factor through the unit of $T$.
A functor $T$ is a parametric right adjoint iff every map $B\to T A$ has a generic factorization.
This is Proposition 2.6 of (Weber08). In fact, the generic factorizations are precisely the universal maps in the left multi-adjoint of $F$ mentioned above.
P.r.a. functors between presheaf categories have an especially nice form.
A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff any map $y(j)\to T 1$ has a generic factorization, where $y(j)$ is the representable presheaf on an object $j\in J$.
This is Proposition 2.10 of (Weber08). The “only if” direction is the previous proposition, while for the “if” direction, the given hypothesis allows us to define the functor
sending an object $(y(j) \to T 1)$ to the object occurring in its generic factorization. Note that $y/T1$ is equivalently the opposite of the category of elements of $T1$. The definition of genericity, along with the Yoneda lemma, then shows that
which preserves connected limits, since it is a coproduct of representables.
In particular, a p.r.a. functor $T\colon [I^{op},Set] \to [J^{op},Set]$ is determined by giving the object $T1\in [J^{op},Set]$ together with the functor $E_T\colon y/T1 = el(T1)^{op} \to [I^{op},Set]$. We can think of $T1(j)$ as the setof all possible “shapes” which $T$ allows us to “glue together” to obtain an element of shape $j$, and $E_T$ as specifying exactly what each of those shapes looks like. Then the above formula for $T(Z)(j)$ says that we look at all possible shapes $x\in T1(j)$ we can glue to get something of shape $j$, and for each such $x$ we look at all the “diagrams” in $Z$ of the corresponding shape $E_T(x)$.
We can extract from this a description that is clearly a generalization of a polynomial functor.
A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff when expressed in terms of discrete fibrations, it is the composite
for a polynomial in $Cat$
where $p$ is a discrete fibration and $(d,c)$ is a two-sided discrete fibration (with in particular $d$ a fibration and $c$ an opfibration).
Let $p$ be the Grothendieck construction of $T1$, so that $K = el(T1)^{op}$, and $(d,c)$ the two-sided Grothendieck construction of $E_T\colon el(T1)^{op} \to [I^{op},Set]$ regarded as a profunctor from $K$ to $I$. The above formula tells us that $T = Lan_p \circ Hom(E_T,-)$, and when rewritten in terms of discrete fibrations this gives the above formula. More details are in Remark 2.12 of (Weber08).
That is, a p.r.a. functor between presheaf categories is the restriction to discrete fibrations of a certain kind of polynomial functor between slices of $Cat$. When $I$ and $J$ are discrete categories, then so are $K$ and $E$, so that p.r.a. functors between presheaf categories are a direct generalization of polynomial functors between slices of $Set$. But on the other hand, we can also say that polynomial functors between slices of Cat are a direct generalization of p.r.a. functors between presheaf categories.
Consider the free category monad $T$ on the category $Quiv$ of quivers, such that $T A$ is the quiver with the same objects as $A$ and whose arrows are finite composable strings of arrows in $A$.. Here $T 1$ is the monoid $\mathbb{N}$ regarded as a one-object category, and thus an object of $Quiv/T1$ is a quiver together with a natural number assigned to each edge. For any quiver $A$, the natural augmentation $T A \to T 1$ assigns to each composable string of arrows its length.
The left adjoint of this functor $T_1\colon Quiv \to Quiv/T1$ takes as input a quiver with natural number “lengths” assigned to each of its arrows, and creates a new quiver by gluing together a copy of the quiver $[n] = (0 \to 1 \to\dots \to n)$ (with no arrows other than those drawn) for each arrow of “length” $n$. Thus $T$ is a parametric right adjoint.
$Quiv$ is of course a presheaf category $[Q^{op},Set]$, where $Q$ is the category $0 \rightrightarrows 1$. The category $y/T1$, i.e. the opposite of the category of elements of $T1$, has objects $\mathbb{N} \sqcup \{\bot\}$ and nonidentity arrows $\bot \rightrightarrows n$ for all $n\in\mathbb{N}$. Finally, the functor $E_T \colon y/T1 \to Quiv$ sends $\bot$ to the quiver with one object and no arrows, and $n$ to the quiver $[n] = (0 \to 1 \to\dots \to n)$ described above.
Aurelio Carboni and Peter Johnstone, Connected limits, familial representability and Artin glueing, MR
Mark Weber, Generic morphisms, parametric representations, and weakly cartesian monads, Theory and applications of categories, 13:191–234, 2004. link
Mark Weber, Familial 2-functors and parametric right adjoints, 2008 link
In database theory p.r.a.s between copresheaf categories, known as data migration functor, are treated in
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