$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and geometric morphisms, where a geometric morphism points in the direction of its inverse image functor. Then $\Logos$ is a non-full subcategory of the $\infty$-category $\Pr^L$ of presentable $\infty$-categories and left adjoint functors.
Question 1: Is the inclusion $\Logos \to \Pr^L$ monadic?
Question 2: If so, is the induced monad lax-idempotent?
I believe this functor preserves limits and filtered colimits. It doesn't preserve coproducts. I'm not sure if it actually has a left adjoint.
If the answer is "yes, up to size issues", that would be interesting too.
I think this might be one of those questions which is cleaner to consider in the $\infty$-categorical context than in the 1-categorical context, but I could be wrong. I'd be interested to hear about the 1-categorical case as well (where I suppose one would consider the $(2,1)$-categories of 1-logoi and locally presentable 1-categories).
Best Answer
Regarding monadicity (rather than comonadicity), the (2-categorical variant of the) question is answered in Bunge–Carboni's The symmetric topos. In their paper, $\mathbf A$ denotes the 2-category of locally presentable categories and cocontinuous functors (i.e. left adjoint functors), and $\mathbf R$ denotes the 2-category of logoi. There is a 2-adjunction $\Sigma : \mathbf A \rightleftarrows \mathbf R : U$ (Theorem 3.1) and the induced 2-monad is colax idempotent (Theorem 4.1 and the following discussion).
Presumably everything works out similarly in the $(\infty, 2)$-categorical setting.