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We exhibit a Gray-category Psm such that a Gray-functor from Psm to a Gray-category A is precisely a pseudomonad in A; this may be viewed as a complete coherence result for pseudomonads. We then describe the pseudoalgebras for a pseudomonad, the morphisms of pseudoalgebras, and so on, as a weighted limit in the sense of Gray-enriched category theory.
We also exhibit a Gray-category Psadj such that a Gray-functor from Psadj to A is precisely a pseudoadjunction in A, show that every pseudoadjunction induces a pseudomonad, and that every pseudomonad is induced by a a pseudoadjunction provided that A admits the limits mentioned in the previous paragraph. Finally we define a Gray-category \psma of pseudomonads in A, show that it contains A as a full reflective subcategory, which is coreflective if and only if A admits these same limits.
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