TITLE: symmetric monoidal double categories?
QUESTION [7 upvotes]: Let me preface this by saying that I don't know much category theory.  
I am running into a situation where I have a double category and additionally there is a multiplication.  Moreover, choosing either the vertical or the horizontal arrows makes my thing a symmetric monoidal category.   Has this structure been studied somewhere?

REPLY [5 votes]: Kenny Courser has written a nice thesis on symmetric monoidal double categories and their applications:

Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020. Available at arXiv:2008.02394.

Also, Mike Shulman's paper mentioned above has a new installment, which explains symmetric monoidal double categories, the maps between them, and the maps between those - and how to turn these into symmetric monoidal double categories, and maps between them, and maps between those:

L. W. Hansen and M. Shulman, Constructing symmetric monoidal bicategories functorially. Available at arXiv:1901.09240.