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This is a continuation of my previous question.

A) Morphisms in (1') are basically internal anafunctors, their compositions heavily use (and only) pullback/limit.

B) Bibundles in (2) are basically special kind internal profunctores, their compositions heavily use quotient/coend/colimit.

We know that (1') and (2) are equivalent, this is a bit surprising since limit and colimit are usually very different notions. My question is which categorical properties make this true? Thanks.

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You should also ask, what categorical properties allow you to DEFINE composition in (2)? Probably this has the same answer. –  David Carchedi Dec 12 '11 at 17:30
@DavidCarchedi I suppose the composition may come from the fact that surjective submersion admits local sections. –  Ma Ming Oct 30 '13 at 17:08
@DavidCarchedi This is addressed in arxiv.org/abs/1408.5220v1 –  Ma Ming Sep 21 '14 at 13:48

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