  For a small $n$-category $\mathscr{C}$ and an $n$-topos $\mathscr{X}$, we
study necessary and sufficient conditions for a functor $f \colon \mathscr{C}
\to \mathscr{X}$ to determine a geometric morphism from $\mathscr{X}$ to the
$n$-topos $\mathcal{P}(\mathscr{C})_n$ of presheaves on $\mathscr{C}$ for any
$n \geq 1$. These results generalize and unify results of Lurie for $n=\infty$
and classical characterizations of flat functors (Diaconescu's theorem) for
$n=1$. Interestingly, for $n=\infty$, our analogue of Diaconescu's theorem
requires hypercompleteness. As an application, we show that the $\infty$-topos
associated to an $n$-site behaves as an $n$-localic $\infty$-topos with respect
to hypercomplete $\infty$-topoi.
