Proof

Following the notation at cohesive topos, we write

for the global section geometric morphism, where the inverse image $\mathrm{Disc}$ constructs discrete objects. We need to exhibit two more adjoints

and show that ${\Pi }_0$ preserves finite products. Finally we need to show that $\Gamma X\to {\Pi }_{0}X$ is an epimorphism for all $X$.

Firstly, since $C$ is a locally connected site, any constant presheaf is a sheaf. This implies that the functor $\mathrm{Disc}$ has a further left adjoint given by taking colimits over $C^{\mathrm{op}}$, which we denote ${\Pi }_0$. Hence $\mathrm{Sh}(C)$ is a locally connected topos.

Moreover, since $C$ is cosifted, ${\Pi }_0$ preserves finite products. In particular, $\mathrm{Sh}(C)$ is connected and even strongly connected.

Next, we claim that $C$ is a local site. This means that its terminal object $*$ is cover-irreducible, i.e. any covering sieve of $*$ must contain its identity map. But since $C$ is a locally connected site, every covering family is inhabited, and since every object has a global section, every covering sieve must include a global section. In the case of $*$, the only global section is an identity map; hence $C$ is a local site, and so $\mathrm{Sh}(C)$ is a local topos. The right adjoint $\mathrm{Codisc}$ of $\Gamma$ is defined by

We now claim that the transformation $\mathrm{Disc}(A)\to \mathrm{Codisc}(A)$ is monic. Since sheaves are closed under limits in presheaves, this condition can be checked pointwise at each object $U\in C$. But since constant presheaves are sheaves, the map $\mathrm{Disc}(A)(U)\to \mathrm{Codisc}(A)(U)$ is just the diagonal

which is monic since $C(*,U)$ is always inhabited (by assumption on $C$).