  Given a morphism of persistence modules (a.k.a. persistence morphism) $f : V
\rightarrow U$, we introduce a novel operator that determines a partial
matching between the barcodes of $V$ and $U$ induced by $f$. We show that the
proposed operator is additive with respect to the direct sum of persistence
morphisms, and that it contains more information than $f V$ and the rank
invariant. We also illustrate some advantages of using our induced partial
matching over the Bauer-Lesnick partial matching $\chi_f$. Lastly, we provide a
family of persistence morphisms that contain modules built from Morse
filtrations, for which $\chi_f$, the rank invariant and our proposed induced
partial matching are equivalent.
