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Asad Saeeduddin (Aug 15 2020 at 05:28):A tensorial strength in Cat looks like this:
This looks remarkably like a naturality square, except that the diagram does not strictly commute. It seems to me that this is the diagram for a lax natural transformation. Is this correct? I.e. is having a tensorial strength for some monoidal structure the same as being a lax natural transformation between the 2-endofunctors and ?
John Baez (Aug 18 2020 at 17:28):I would guess you're right, but there's a way to test your hypothesis. If the 2-morphism str is part of a lax natural transformation, it needs to obey some equations, which you can see here. (Leinster calls it a "transformation", and the equations look complicated since he's working in a bicategory, so there are associators and left/right unitors running around, which you can ignore since they're identities in Cat.) So, you can write down these equations and see if they're part of the definition of "tensorial strength".
It's fun to do these exercises since when you succeed you feel a sense of triumph.