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With reference to the terminology “monoidal category”, I think this ought to be called a “magmoidal category” (this appears to have been used more frequently in practice too).
Changed name as suggested. I agree, it goes better with monoidal category, which is the categorification of a monoid, and appears in the literature: Davydov’s Nuclei of categories with tensor products and Lack and Street’s Triangulations, orientals, and skew monoidal categories, and honourable mention to H. Eades III in Introducing a New Project on The Combination of SubstructuralLogics and Dependent Type Theory (extended abstract).
Added example of Ackermann groupoids.
However, I do wonder if one should ask that $\rho_I = \lambda_I$…
I agree that one should: as a term rewriting system, these rewrites should be convergent, so any choice of path should be equal.
Added note on a potentially necessary weak coherence condition for magmoidal categories with units, namely that there should be a unique coherence iso $I\otimes I \to I$, so the unitors should coincide. Also noted the usual unitor coherence condition can’t even be stated in the absence of associators.
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