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I would like to draw your attention to this thread I recently opened in MO: https://mathoverflow.net/questions/285872/special-gamma-categories-and-symmetric-monoidal-categories It should be relevant around here.
In particular, note that the “symmetric monoidal category” page of the nLab does not have a section on the “unbiased” versions.
(Added the link – thanks, David, for showing me how to do it)
https://mathoverflow.net/questions/285872/special-gamma-categories-and-symmetric-monoidal-categories (click “Source” at top right of this comment to see how I did this. The format “Format comments as” button is set to Markdown+Itex)
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