@DylanWilson I can't seem to find/discern what is the "invariant" statement there i.e. what is the straightning of a locally co-cartesian fibration? Is it a lax functor to the $2$-category of $\infty$-cateogries like Alexander said or is it something else perhaps?

We say a topos has enough points if it admits a jointly conservative family of left exact cocontinuous functors into the category of sets.

Is there some terminology for the analogue of this condition for Grothendieck abelian categories? That is, how would we call a Grothendieck abelian category which admits a conservative, cocontinuous exact functor into abelian groups?

@AaronMazel-Gee but it is! My comment was a correction to your (2) not your (1)

@SaalHardali To be precise they correspond to normal lax functors (where normal = preserves identities), which can be defined in the scaled simplicial set model as maps out of a quasicategory with the minimal scaling

Won’t they be “normal oplax” in the locally cocartesian case? I always get these confused but it should be whichever word means the arrow goes F(f\circ g)—->F(f) \circ F(g), right?

Probably, I usually use lax as shorthand for "whichever of lax/oplax/colax is correct" :-)

@RuneHaugseng Thanks