  For a small quantaloid $\mathcal{Q}$ we consider four fundamental 2-monads
$\mathbb{T}$ on $\mathcal{Q}\text{-}{\bf Cat}$, given by the presheaf 2-monad
$\mathbb{P}$ and the copresheaf 2-monad $\mathbb{P}^{\dagger}$, as well as by
their two composite 2-monads, and establish that they all laxly distribute over
$\mathbb{P}$. These four 2-monads therefore admit lax extensions to the
category $\mathcal{Q}\text{-}{\bf Dist}$ of $\mathcal{Q}$-categories and their
distributors. We characterize the corresponding
$(\mathbb{T},\mathcal{Q})$-categories in each of the four cases, leading us to
both known and novel categorical structures.
