Let $\mathcal C$ be a sufficiently nice $\infty$-category, and let $f: U \to X$ be a morphism in $\mathcal C$. Recall that $f$ is said to be an effective epimorphism if the induced map $|U^{\times_X (\bullet+1)}| \to X$ is an equivalence, where we have taken the geometric realization of the Cech nerve of $f$, whose $n$th level is $U^{\times_X (n+1)} = U\times_X \dots \times_X U$.
Question 1: Let $\mathcal C = Pr^R$ be the $\infty$-category of presentable $\infty$-categories and right adjoint functors. If $f_\ast: \mathcal U \to \mathcal X$ is a morphism in $Pr^R$, is there a more elementary characterization of when $f_\ast$ is an effective epimorphism in $Pr^R$?
Since limits in both $Pr^R$ and the opposite category $Pr^L$ are computed at the level of underlying categories, we can at least say that the $n$th level of the Cech nerve is $\mathcal U^{\times_{\mathcal X}(n+1)} = \mathcal U \times_{\mathcal X} \dots \times_{\mathcal X} \mathcal U$ where the pullback is taken in $Cat$ over the functor $f_\ast$; its geometric realization in $Pr^R$ is the totalization in $Cat$ of the cosimplicial category with those levels, whose cosimplicial coface maps are given by the left adjoints of the various projections. But it seems tricky to describe these left adjoints explicitly, which is hindering the analysis for me.
The one case I think I understand is when $f_\ast$ is a colocalization, i.e. its left adjoint $f^\ast$ is fully faithful. In this case, I believe the left adjoints to the projections $\mathcal U \times_{\mathcal X} \mathcal U ^\to_\to \mathcal U$ are given by $U \mapsto (U, f^\ast f_\ast U)$ and $U \mapsto (f^\ast f_\ast U, U)$, and I think it looks like $f_\ast$ does indeed turn out to be an effective epimorphism.
Question 2: Are colocalizations examples of effective epimorphisms in $Pr^R$?
Even if so, I'd be hesitant to guess that these are all the examples there are.
Question 3: A natural guess is that $f_\ast$ is is an effective epimorphism in $Pr^R$ if and only if its left adjoint $f^\ast$ is conservative. Is this correct?
I'd be interested in conclusions which make further assumptions about $\mathcal U, \mathcal X$ -- for instance, I'd be happy to assume they're stable. I believe the inclusion from stable presentable $\infty$-categories to all presentable $\infty$-categories preserves limits and colimits, so the geometric realization of the Cech nerve is computed in the same way, so that this really would amount to a special case.
EDIT: What I'd ultimately like to know is what are the effective epimorphisms in the category $(E_\infty(Pr^L))^{op}$ of symmetric monoidal presentable $\infty$-categories, with the morphisms being lax monoidal right adjoints whose left adjoints are strong monoidal. I've just realized that since pullbacks in this category are computed differently, the question as I've asked it above is not going to tell me the answer to this question. So I'll ask:
Question 4: What are the effective epimorphisms of the category $(E_\infty(Pr^L))^{op}$ of symmetric monoidal presentable $\infty$-categories? How about in the stable case?
For good measure, it would be interesting to know what the effective epimorphisms of $\infty$-topoi are. Colimits in $Topoi^R$ are again computed by taking limits of left adjoints, but limits are not generally computed by taking limits of right adjoints, so the Cech nerve probably looks different.
