Cospans of finite sets

Cospans of finite sets are the morphisms in a bicategory, not really a category, because composition of cospans is associative only up to natural isomorphism. How can we characterize this bicategory abstractly? There's a category of finite sets and isomorphism classes of cospans, and Steve Lack gave a beautiful characterization of this, which I will explain. But what about the bicategory? I will state a guess that I haven't proved.