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Does anyone know a book or article in a journal that defines biprofunctors between bicategories $\Bbb A$ and $\Bbb B$, as in nLab, i.e., as a pseudo functor $\Bbb A^{op}\times\Bbb B\to\Bbb C\rm{at}$? Perhaps it also exists under some other name, that I'm not familiar with..

I prefer not to refer only the nLab webpage..

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It looks like this is part of Finn Lawler's thesis outline. Did you contact him? –  Martin Jan 13 '13 at 18:08
    
Not yet. Good idea, thanks. But, perhaps it also exists under some other name. –  Berci Jan 13 '13 at 18:22
    
In my (still!) unfinished PhD thesis, I used to call your "biprofunctors" Cat-valued profunctors/distributors/modules, or just Cat-profunctors. Another closely related concept is "2-discrete fibred span". –  Michal R. Przybylek Jan 13 '13 at 18:42
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