> In your perception, tensoring x composing (and or series x parallel) could be framed as related to duality?
I said no in comment #10 when you asked that in comment #9, and I'll say it again: no.
> You recently presented "Profunctor theory is to category theory as linear algebra is to set theory!". Is there some other analogies like this that you could present for us?
There are tons of analogies! My brain is packed with analogies. For example, in my previous two comments I explained the analogy between profunctors and relations, and the analogy between relations and matrices. Putting them together gives the analogy between profunctors and matrices, which is the reason profunctor theory is like linear algebra.
Of course, these analogies are only useful if you know exactly how they work and also how they _don't_ work.
Every really good analogy is yearning to be a functor; when it's a functor we can work with it mathematically in a rigorous way.