Does finiteness imply cancellativity in semiadditive CD categories?

Determine whether there exists a semiadditive copy-discard (CD) category in which a finite morphism fails to be cancellative; that is, ascertain if the implication "finite morphism implies cancellative" does not hold in general by constructing a counterexample.

Background

The paper introduces semiadditive CD categories, which are CD categories enriched over commutative monoids, allowing sums of morphisms and a preorder on hom-sets. Within this setting, the authors define finiteness of a morphism via cancellativity of the effect 1∘P and show, in categories of kernels (Kern and sfKern), that finiteness implies cancellativity, corresponding to pointwise σ-finiteness.

Motivated by whether this implication holds beyond classical kernel categories, the authors explicitly raise doubt that finiteness implies cancellativity in arbitrary semiadditive CD categories and introduce the notion of "finitely cancellative" to capture cases where the implication does hold.

References

It is therefore natural to ask whether this relationship holds generally. We conjecture that it does not always hold, and so introduce the following definition to capture the cases where it does.

— A categorical account of the Metropolis-Hastings algorithm  (2601.22911 - Cornish et al., 30 Jan 2026) in Section 4.2 (Cancellative and finite morphisms)