Author’s abstract: “We define distributive laws between pseudomonads in a Gray-category \({\mathcal A}\), as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the Gray-category \(\text{PSM} ({\mathcal A})\) of pseudomonads in \({\mathcal A}\), and define a lifting to be a pseudomonad in \(\text{PSM} ({\mathcal A})\). We define what is a pseudomonad with compatible structure with respect to two given pseudomonads. We show how to obtain a pseudomonad with compatible structure from a distributive law, how to get a lifting from a pseudomonad with compatible structure, and how to obtain a distributive law from a lifting. We show that one triangle suffices to define a distributive law in case that one of the pseudomonads is a (co-)KZ-doctrine and the other a KZ-doctrine”.