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fosco (Dec 16 2020 at 20:11):This is admittedly crossposting; if I am violating any rule just tell me! The only reason I am throwing a link like this is that zulip chat does not support AMScd diagrams...
Yesterday I posted on MO this thread https://mathoverflow.net/questions/378972/what-is-a-bipullback-of-lax-functors
This is the only missing spot in a bigger question. I think I'm really this far from proving it, and what I've said is mostly correct, but -even after all this time- I lack command of the 2-dimensional subtleties.