Somehow I still have this working hypothesis that a weak n-category is just a certain funny way of looking at a strong n-category; and I still don't know how to reconcile this with the fact that it's known that things break down at 3 dimensions, that every tricategory is triequivalent to a Gray-category, but not a strong 3-category. The simple explanation of this failure — that a strong 3-category with one object and one morphism plainly has a strict symmetry, whereas a tricategory with one object and one morphism has merely a braiding — doesn't satisfy much since it seems the right thing to compare the fully weakened situation to so a strong 3-category where all objects are suitably equivalent and likewise each pair of morphisms within any given homset is equivalent.
But skimming through TWF again I found that there's a nice string-diagram explanation for how matrix multiplication arises as a monadic adjunction in the monoidal category of vector spaces and tensor products.
Also: to get a monad from an adjunction, the triangle equalities seem to only be used to establish the unit laws of the monad, not the multiplication associativity.
I have this sinking feeling if I stopped doing legitimate, more or less coherently reasoned mathematics and completely went off the deep end and made livejournal entries full of pure gibberish, few people would be able to tell the difference.