I have two questions on the 2-category of prederivators $\bf PDer$:

Does $\bf PDer$ "admit the construction of algebras" (ATCOA) in the sense of Street's Formal theory of monads? I recall that by ATCOA for a 2-category $\cal K$ I mean that the inclusion functor ${\cal K} \to {\bf Mnd}({\cal K})$ has a right 2-adjoint. One can show by hand (see below) that there are well-behaved objects of algebras for a monad $T : \mathbb D \to \mathbb D$, but I wouldn't be able to guess if something goes wrong when trying to glue these objects to a 2-adjoint ${\bf Mnd}({\bf PDer})\to \bf PDer$...

Regarding a monad as a lax functor $1\to \cal K$ its "object of EM-algebras" is its lax limit (dually, the Kleisli object is its lax colimit). Is it known that given a monad $T$ on a prederivator $\mathbb D$ there are adjunctions $$ \text{Alg}_T(\mathbb D) \leftrightarrows \mathbb D \quad\text{and}\quad \text{Kl}_T(\mathbb D) \leftrightarrows \mathbb D $$ obtained gluing the various categories of algebras and free algebras $\text{Alg}_{T_J}(\mathbb D(J))$ and $\text{Kl}_{T_J}(\mathbb D(J))$ on the monads $T_J : \mathbb D(J) \to \mathbb D(J)$ induced by the components of $T$, in order to form the prederivator of EM-algebras and of free algebras.

Are these two prederivators still characterized by the universal property of the lax co/limit of $T : 1\to \bf PDer$?