  We show that $\mathit{ann}$-categories admit a presentation by crossed
bimodules, and prove that morphisms between them can be expressed by special
kinds spans between the presentations. More precisely, we prove the groupoid of
morphisms between two $\mathit{ann}$-categories is equivalent to that of
bimodule butterflies between the presentations. A bimodule butterfly is a
specialization of a butterfly, i.e. a special kind of span or fraction, between
the underlying complexes
