Bénabou's theorem for pseudoadjunctions

Abstract: We give a formal account of B\'enabou's theorem for peudoadjunctions in the context of Gray-categories. We prove that to give a pseudoadjunction $F \dashv U: A \to X$ with unit $\eta$ in a Gray-category K is precisely to give an absolute left (Kan) pseudoextension $U$ of $1_X$ along $F$ witnessed by $\eta$.