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Wrote a bit at double profunctor.
Hi unmuamua, just to say that it’s great that you posted your question here, and that you followed the discussion link to here; that is exactly the intended purpose! I’m sure Mike or someone else knowledgeable will take a look at your question soon.
Hi! There are no other references that I know of; that page is original research. (To link to an nLab page in an nForum post, put its name in double square brackets, like [[double profunctor]]
.) The two subsections under “Composition” describe two ways to define composition, so composites of a sort certainly do “exist”. The ill-behavedness referred to is more explicitly that neither of the two composites mentioned satisfies the necessary conditions to be associative. I haven’t worked out a specific example of double profunctors $H,K,L$ such that $(H\circ K)\circ L \neq H\circ (K\circ L)$, but you could probably do it as an exercise by starting with the relevant input counterexample (either of a coequalizer in $Cat$ not preserved by pullback or a bo-ff factorization in $DblCat$ not preserved by pushout). If you do, please add it to the nLab page!
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