  This paper investigates some issues arising in categorical models of
reversible logic and computation. Our claim is that the structural (coherence)
isomorphisms of these categorical models, although generally overlooked, have
decidedly non-trivial computational content. The theory of categorical
coherence is based around reversible structural operations (canonical
isomorphisms) that allow for transformations between related, but distinct,
mathematical structures. A number of coherence theorems are commonly used to
treat these transformations as though they are identity maps, from which point
onwards they play no part in computational models. We simply wish to point out
that doing so overlooks some significant computational content.
  We give a single example (taken from an uncountably infinite set of similar
examples, and based on structures used in models of reversible logic and
computation) of a category whose structural isomorphisms manipulate modulo
classes of natural numbers. We demonstrate that the coherence properties that
usually allow us to ignore these structural isomorphisms in fact correspond to
countably infinite families of non-trivial identities in modular arithmetic.
Further, proving the correctness of these equalities without recourse to the
theory of categorical coherence appears to be a hard task.
