Concerning the question by Lindquist: The tensor product automatically satisfies all functoriality, associativity and coherence conditions, if it is introduced by a universal property as by Bourbaki. This is shown for monoidal categories (bicategories with one object) e.g. in my paper ``Multicategories revisited'', Contemporary Mathematics 92(1989). The same argument works for arbitrary bicategories provided, in defining a multicategory, one replaces the free monoid generated by a set by the free category generated by a graph. Jim Lambek
From: "Prof. J. Lambek" <lambek@math.mcgill.ca> Subject: categories: Reading advice
Concerning the question by Lindquist:
The tensor product automatically satisfies all functoriality, associativity and coherence conditions, if it is introduced by a universal property as by Bourbaki.
In view of this would it be fair to say that coherence is not a notion intrinsic to category theory, but rather arises from the traditional set theoretic presentation (or at least point of view) of category theory? Much the same can surely be said of naturality, whose abstract essence is that of 2-cells but which is standardly presented concretely, where the interchange axiom becomes a not entirely trivial theorem. Vaughan Pratt
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Prof. J. Lambek -
Vaughan Pratt