A strongly connected site is a site satisfying sufficient conditions to make its topos of sheaves into a strongly connected topos.
Let $C$ be a locally connected site; we say it is a strongly connected site if it is also a cosifted category
If $C$ is strongly connected site, then the sheaf topos $Sh(C)$ is a strongly connected topos.
Because the left adjoint $\Pi_0$ in the sheaf topos over a locally connected site is given by the colimit functor and colimits preserve finite products on the sifted category $C^{op}$.
and
locally connected site / locally ∞-connected site
strongly connected site / strongly ∞-connected site