I have a question that has been puzzling me for some days: let $\mathcal{O}\colon \mathbf{Cat}_{\infty} \to \mathbf{Set}$ be the composite $\pi_0 \circ k$, where $k$ is the maximal subgroupoid functor. I believe I can show that the maps inverted by $\mathcal{O}$ correspond to the strong saturation of $\partial \Delta^1 \to \Delta^1$, and I am wondering whether fully faithful functors are precisely those functor with the right lifting property with respect to these maps.

In other words, I am looking for a sort of (bijective on objects, fully faithful) factorization system, possibly originating from a cartesian fibration $\mathbf{Cat}_{\infty}\to \mathbf{Set}$.

@TimCampion Here's something slightly more precise. Let $f_d: M_d \to N_d$ be a natural transformation of diagrams which is pointwise a homotopy equivalence. Consider the functor on the twisted arrow category of $D$ which assigns to $g: c \to d$, the space of maps \{$h: N_c \to M_d$ such that $f_d \circ h = M_g$\} (really, you want a space of maps and homotopies).

Then I think this space is contractible, and the space of inverse natural transformations is given by taking the limit over $Tw(D)$. So you just need that a limit of contractible spaces is contractible.

@EdoardoLanari The functors left orthogonal to the fully faithful functors should be the essentially surjective on objects functors, i.e. those which are surjective on $\pi_0 \circ k$.

What kind of limits/colimits exist in $Pr^{L, \kappa}$ (functors are colimit preserving and preserve $\kappa$-compact objects) and are also preserved by the inclusion to $Pr^L$?

Goodwillie calculus

Sorry, wanted to search this instead of sending