This question is answered in the affirmative in the following preprint:
- Cole Comfort, Antonin Delpeuch, Jules Hedges, Sheet diagrams for bimonoidal categories, arXiv:2010.13361
The morphisms are represented by so-called proof sheets.
Here is an example of such a proof sheet, representing the controlled-not gate using the program sheetshow which Antonin created to typeset the diagrams in the preprint:
