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Thanks to @slibkind's PR #433, we now have a first implementation of the left pushforward data migration functor for C-sets. Given a functor F: C -> D between schemas, the algorithm assumes that the graph generating D has no cycles (it performs a topological sort on D). This means that we cannot, for example, create the free reflexive graph on a graph because the schema for reflexive graphs has cycles.
In general, computing left Kan extensions is a challenging problem. The state of the art seems to be Fast Left Kan Extensions using the Chase by Spivak and Wisnesky. For a next step, I think we should just do the simplest thing that would cover standard examples like symmetric and/or reflexive graphs.