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I am a bit confused, I always thought that the canonical setup for pseudomonoids were Gray-categories hence this should also hold for their horizontal categorification. Where am I wrong ?
Re. 3: It’s the same distinction as in the one-dimensional setting, where one defines monoids internal to any monoidal category, but categories internal to any category with pullbacks. There’s a generalisation of the latter to monoidal categories with nice equalisers: perhaps that is possible also for pseudocategories, but I don’t believe anyone has done so.
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