  This paper is concerned with categorical structures for reversible
computation. In particular, we focus on a typed, functional reversible language
based on Theseus. We discuss how join inverse rig categories do not in general
capture pattern-matching, the core construct Theseus uses to enforce
reversibility. We then derive a categorical structure to add to join inverse
rig categories in order to capture pattern-matching. We show how such a
structure makes an adequate model for reversible pattern-matching.
