Indiscrete objects in a functor category

[Aleks Kissinger <aleks0@gmail.com>]

It's common to describe the category of (directed, multi-) graphs as a
functor category Graph := [2, Set], where 2 here is the category with
2 objects and 2 parallel arrows (s & t).

For a pair of sets (V,E), one can construct the indiscrete graph
I(V,E) as a graph with vertices V and edges E x V x V, where the
source and target maps are just the 2nd and 3rd projection
respectively. This gives a right adjoint to the forgetful functor from
Graph to pairs of sets. This enables one to construct a category of
graphs with a fixed set of vetex/edge labels as a slice over Graph:

Graph / I(Lv, Le)

since a graph hm G --> I(Lv,Le) is the same as a map U(G) --> (Lv,Le),
which is just a pair of functions assigning labels to the vertices and
edges of G.


This seems to me like a pretty standard trick, which extends to any
functor category from a C which is in some sense "suitably acyclic".
For instance, consider a category of "partitioned graphs" [3, Set],
where 3 has objects (P,V,E) and arrows:

E --s--> V, E --t--> V, and V --p--> P

where, p assigns each of the vertices a partition. For a triple
(P,V,E) we can form the indiscrete partitioned graph with:

- partitions P
- vertices V x P
- edges E x (V x P) x (V x P)
- p = pi2, s = pi2, t = pi3

which gives a right-adjoint to the forgetful functor from partitioned
graphs to triples of sets. This is clearly an instance of a general
recipe, whereby you start with the objects with no arrows out, and
work your way backwards, always adding copies of the codomain of every
out-arrow. Again one can attach labels to partitioned graphs by
slicing:

[3,Set] / I(Lp,Lv,Le)


So, my question: Is the general case a known/studied construction? If
so, could someone provide a reference?



Best,

Aleks


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