  We build a functorial pipeline for persistent homology. The input to this
pipeline is a filtered simplicial complex indexed by any finite metric lattice
and the output is a persistence diagram defined as the M\"obius inversion of
its birth-death function. We adapt the Reeb graph edit distance to each of our
categories and prove that both functors in our pipeline are $1$-Lipschitz
making our pipeline stable. Our constructions generalize the classical
persistence diagram and, in this setting, the bottleneck distance is strongly
equivalent to the edit distance.
