Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty Sh(C, J)$. There is a natural geometric morphism to $\infty \text{Grpd}$ whose left adjoint is the constant sheaf functor $\Delta : \infty \text{Grpd} \to \infty Sh(C, J)$.
In the $1$-topos case, $\Delta$ acts as follows: Take a set $S$ and take the constant presheaf on $S$, then sheafify. The resulting sheaf has the property that $\Delta(S)(U) = S$ for $U$ connected. Is this also true in the $\infty$ case, or do we need $U$ to be contractible? What even is $\Delta$??