The category of commutative monoid objects in a symmetric monoidal category is cocartesian, with their tensor product serving as their coproduct. This sort of result seems to date back to here:


* Thomas Fox, Coalgebras and Cartesian categories, *Commun. Algebra* **4** (1976), 665–667.


I'm working on a paper with Todd Trimble and Joe Moeller, and right now we need something similar one level up — that is, for [symmetric pseudomonoids](https://ncatlab.org/nlab/show/pseudomonoid). (For example, a symmetric pseudomonoid in Cat is a symmetric monoidal category.)


The 2-category of symmetric pseudomonoids in a symmetric monoidal 2-category should be cocartesian, with their tensor product serving as their coproduct. I imagine the coproduct universal property will hold only up to 2-iso.


Has someone proved this already? This paper:


* Brendan Fong and David I, Spivak, [Supplying bells and whistles in symmetric monoidal categories](https://arxiv.org/abs/1908.02633).


proves the result in the special case where the symmetric monoidal 2-category is Cat. In fact they do more, in this special case:


**Theorem 2.3.** The 2-category SMC of symmetric monoidal categories, strong monoidal functors, and monoidal natural transformations has 2-categorical biproducts.


Unfortunately their proof is not purely 'formal', so it doesn't instantly generalize to other symmetric monoidal 2-categories. And I believe the fact that the coproducts in SMC are *biproducts* must rely on the fact that Cat is a *cartesian* 2-category.



Is the tensor product of symmetric pseudomonoids their coproduct?
The result I wanted is Theorem 5.2 here:


Daniel Schäppi, [Ind-abelian categories and quasi-coherent sheaves](https://arxiv.org/abs/1211.3678), *Mathematical Proceedings of the Cambridge Philosophical Society*, **157** (2014), 391–423. [doi:10.1017/S0305004114000401](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/indabelian-categories-and-quasicoherent-sheaves/21B0BB7544A3D26255324F36FA3DA712)


The proof appears in Appendix A. He proves the result for symmetric pseudomonoids in a symmetric monoidal bicategory.


