A locally ∞-connected (∞,1)-topos
is called strongly connected if $\Pi$ preserves finite (∞,1)-products (hence in particular the terminal object, which makes it also an ∞-connected (∞,1)-topos).
Similarly for an $n$-connected $(n,1)$-topos.
For $n = 1$ this yields the notion of strongly connected topos.
If in addition $\mathbf{H}$ is a local (∞,1)-topos then it is a cohesive (∞,1)-topos.
locally connected topos / locally ∞-connected (∞,1)-topos
strongly connected topos / strongly $\infty$-connected $(\infty,1)$-topos
and