I am working with enriched directed graphs (aka, directed graphs/quivers such that the edges are objects in a category V). I can write every graph as a filtered colimit of finite graphs, and I can write every edge as a filtered colimit of a certain kind of objects in V.
Is there a way to combine the two in order to write every graph as a filtered colimit of finite graphs whose edges are those objects in V? My intuition says yes, but I can't see how to formalise that.
(I tried to phrase this in a more general way, but the category V is cochain complexes over a ring and the "certain objects" are perfect complexes, if that helps.)
Edit: "Enriched directed graphs" are given by a set of objects $Obj(X)$ and a collection of edges $X(x,y)_{x,y\in Obj(X)}$ that are all objects in $V$.