The formal theory of monads on $\bf PDer$

Question

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

1. 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 ${\bf Mnd}({\bf PDer})\to \bf PDer$...

2. 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 $T : 1\to \bf PDer$?

Answer 1

The 2-category of prederivators is just the functor 2-category $[\mathrm{Dia}^{op},\mathrm{Cat}]_p$ with pseudonatural transformations as morphisms. As long as the domain 2-category $\mathrm{Dia}$ is small with respect to the codomain $\mathrm{Cat}$, this is of the form $T$-$\mathrm{Alg}_p$, i.e. the 2-category of strict $T$-algebras and pseudo $T$-morphisms, for a suitable 2-monad on the 2-category $\mathrm{Cat}^{\mathrm{ob}(\mathrm{Dia})}$. Blackwell-Kelly-Power "Two-dimensional monad theory" showed that for a good 2-monad $T$, the 2-category $T$-$\mathrm{Alg}_p$ has pie-limits lifted from the underlying 2-category, which include lax limits such as EM-objects, and also has bicolimits by a more involved construction, which include lax bicolimits such as "weak Kleisli objects".

So the answer to question (1) is yes, and the answer to question (2) is yes for the EM-object with the usual strict universal property (an isomorphism of hom-categories), whereas a "Kleisli object" with a weak universal property (an equivalence of hom-categories) does exist but is not a priori formed objectwise. I would expect it to be "objectwise up to equivalence", since "bicolimits in functor bicategories are objectwise" ought to be true, but I don't know an easy proof of that offhand.

Edit: Alexander Campbell points out that actually strict Kleisli objects exist as well, since $[\mathrm{Dia}^{op},\mathrm{Cat}]_p$ is also 2-comonadic.