$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$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).