A pseudoadjunction is a kind of weak 2-adjunction. Sometimes the word is used interchangeably with biadjunction to mean a “fully weak” notion of 2-dimensional adjunction. Other times, a “pseudoadjunction” refers to the weakest kind of 2-adjunction definable in a Gray-category, which when specialized to the prototypical Gray-category $Gray$ consists of strict 2-categories and strict 2-functors, but pseudonatural transformations.
Steve Lack, A coherent approach to pseudomonads
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