A monad (= triple) on an object \(A\) consists of an endomorphism \(t\) on \(A\) together with multiplication and unit 2-cells which satisfy associativity and unital equalities. This notion makes sense in any 2-category (or even any bicategory). A pseudomonad has the same data plus coherent invertible 3-cells replacing the equalities. This notion makes sense in any tricategory; however, in view of a coherence result of R. Gordon, A. J. Power and R. Street [“Coherence for tricategories”, Mem. Am. Math. Soc. 558 (1995; Zbl 0836.18001)], the author works in a Gray-category rather than a general tricategory. It is classical that a monad gives rise to a co-augmented cosimplicial object of \(\text{End}(A)\); in fact, a strict monoidal functor from the monoidal category \({\mathcal O}rd\) of finite ordinals to \(\text{End}(A)\). The author constructs a Gray-monoid \({\mathcal O}rd\)’ such that a pseudomonad amounts to a Gray-monoid morphism from \({\mathcal O}rd\)’ to \(\text{End} (A)\). Rewriting techniques are used to prove this. The paper then discusses algebras for pseudomonads and other aspects of the “formal theory of pseudomonads”. This paves the way for an alternative approach to F. Marmolejo [“Distributive laws for pseudomonads”, Theory Appl. Categ. 5, 91-147 (1999; Zbl 0919.18004)].