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this kind of entry has been missing: On (versions of) the 2-category of categories with adjoint functors between them.
Starting here some minimum, for the moment just copying the paragraphs that I just added at transformation of adjoints
In the infinity category case, Higher Topos Theory has the basic elements, although it only spells things out in the case of presentable fibrations. I packaged them together at https://ncatlab.org/nlab/show/adjoint+(infinity%2C1)-functor#category_of_adjunctions . (the term “adjunct fibration” is my own)
I added a link to the related concepts.
Incidentally, in HTT, Lurie uses “bifibration” to refer to fibrations over $C \times D$ corresponding, under a two-variable version of the Grothendieck construction, to bifunctors $C^{op} \times D \to \infty Gpd$, if I’ve worked through everything correctly. So it may be confusing to import similar language over to that article.
Lurie uses “bifibration” to refer to fibrations over $C \times D$
This is related to the point that BryceClark just made in another thread: here. For these structures we should probably point to two-sided fibration,
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