Before revision:  Generalizing this idea allows freedom of choosing the indexing poset P of F: P \rightarrow C in defining Patel's } generalized persistence diagram by A. ~Patel to zigzag modules valued in a symmetric monoidal bicomplete category .
After revision:  Generalizing this idea allows freedom of choosing the indexing poset P of F: P \rightarrow C in defining Patel's } generalized persistence diagram of F. By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type \mathcal{A .