(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
A strongly $\infty$-connected site is a site satisfying sufficient conditions to make the (∞,1)-sheaf (∞,1)-topos over it a strongly ∞-connected (∞,1)-topos.
Let $C$ be a ∞-connected locally ∞-connected site; we say it is a strongly $\infty$-connected site if it is also a cosifted (∞,1)-category.
If $C$ is in addition an ∞-local site then it is an ∞-cohesive site.
If $C$ is a strongly $\infty$-connected site, then the (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(C)$ over it is a strongly ∞-connected (∞,1)-topos.
We need to check that the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos-functor $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves finite (∞,1)-products.
By the discussion at ∞-connected site we have that $\Pi$ is given by the (∞,1)-colimit (∞,1)-functor $\lim_\to : Func(C^{op}, \infty Grpd) \to \infty Grpd$. On the opposite and therefore sifted (∞,1)-category $C^{op}$ these preserve finite (∞,1)-products.
and
locally connected site / locally ∞-connected site
strongly connected site / strongly ∞-connected site