The paper has a lot more. It includes many of the basic things that go along with tensor products, and the details of how they work in this setting. If you're really into that kind of thing, I think you'll find it useful!

Here, I just want to add a couple of weird/neat things: (1) coproducts via the Gray tensor product; (2) subtleties about the monoidal unit.

(1) The direct sum (coproduct) of permutative categories comes from the Gray tensor product! For permutative categories A and B, you can form one-object 2-categories ΣA and ΣB. Then their Gray tensor product, ΣA ⊠ ΣB, is another 1-object 2-category, and it has a permutative category C of 1- and 2-cells.

That permutative category C is the direct sum A ⊕ B. We also give a straightforward objects-and-morphisms definition, but I think the connection with the Gray tensor product is neat. This direct sum is equivalent (not isomorphic) to the cartesian product!

(2) The monoidal unit, S, is also known as the natural number category. It's objects are natural numbers, and it's morphisms are given by permutations of n-elements sets. A subtle but apparently important fact is that the 2-unitors are not strict.

This seems related to corresponding monoidal unit subtleties for (a) EKMM S-modules, where S is the sphere spectrum, and (b) Elmendorf-Mandell M1-modules (in pointed multicategories). We don't have a mathematical statement of this relationship, just a feeling we can't put into words.

Thanks for reading! If you want to know more, go check that paper out. It's long, but the details are relatively straightforward. We tried to organize it so thoroughly that people can find what interests them with relative ease. An appendix about symmetric monoidal bicategories marks all the data that are trivial (or not) for permutative categories! A coherence theorem makes checking the nontrivial part pretty easy.

Here's what I was trying to convey about the diagram looking like a crab.