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There is a nice result for regular 1-categories, which I quote from page 441 of Borceux & Bourn's textbook "Mal'cev, Protomodular, Homological and Semi-Abelian Categories". This is ...

Let $(\mathcal{C},\otimes)$ be a symmetric monoidal bicategory. Assume that $\mathcal{C}$ has bicategorical coequalizers which are preserved by $\otimes$ in each variable. My question is if then the ...