Pseudo-commutativity of KZ 2-monads

In this paper we prove that KZ 2-monads (also known as lax-idempotent 2-monads) are pseudo-commutative. The main examples of KZ 2-monads for us will be 2-monads whose algebras are V-categories with chosen colimits of a given class; this provides a large family of examples of pseudo-commutative 2-monads. In order to achieve this we characterise pseudo-commutativities on a 2-monad in terms of extra structure on its 2-category of algebras and pseudomorphisms. We also consider tensor products associated to pseudo-closed structures and show some results on preservation of colimits. To cover the general case of V-enriched categories and not only ordinary categories we are led to consider monads enriched in a 2-category, and some of the associated two-dimensional monad theory.