Let $P$ be a distributor/profunctor from a small category $A$ to a small category $B$, i.e. a functor $P : B^\circ \times A \to \mathrm{Set}$.
We may then define a distributor from $[A^\circ, \mathrm{Set}]$ to $[B^\circ, \mathrm{Set}]$, i.e. a functor $\hat P : [B^\circ, \mathrm{Set}]^\circ \times [A^\circ, \mathrm{Set}] \to \mathrm{Set}$, as follows. It suffices to construct a functor $\tilde P : (\mathrm{DFib}/B)^\circ \times \mathrm{DFib}/A \to \mathrm{Set}$, which then corresponds to $\hat P$, since $[(-)^\circ, \mathrm{Set}] \simeq \mathrm{DFib}/(-)$ via the category of elements construction. $$ \tilde P(p, q) := \mathrm{lim}_{i \in I} \mathrm{colim}_{j \in J} P(p(i), q(j)) $$
In particular, when $P$ is the hom-set functor $A(-, -) : A^\circ \times A \to \mathrm{Set}$, the distributor $\hat P$ is isomorphic to the identity distributor on the presheaf category $[A^\circ, \mathrm{Set}]$, whose morphisms may similarly be characterised via a limit–colimit formula (see Proposition 3.11 of Presentations of clusters and strict free-cocompletions, for instance).
This construction seems very natural. Does it, or any reminscent construction, appeared in the literature?
