Proof

On the one hand, if $p^{*}$ is fully faithful, then the counit $p_{!}{p}^{*}\to Id$ is an isomorphism, so we have $p_{!}(*)\cong p_{!}(p^{*}(*))\cong *$; hence $p_{!}$ preserves the terminal object.

On the other hand, suppose that $p_{!}$ preserves the terminal object. Suppose also for simplicity that $F=\mathrm{Set}$. Then any set $A$ is the coproduct ${\coprod }_A*$ of $A$ copies of the terminal object. But $p^{*}$ and $p_{!}$ both preserve coproducts (since they are left adjoints) and terminal objects (since $p^{*}$ is left exact, and by assumption for $p_{!}$), so we have

Thus, the counit $p_{!}{p}^{*}\to Id$ is an isomorphism, so $p^{*}$ is fully faithful.

When $F$ is not $\mathrm{Set}$, we just have to replace ordinary coproducts with ”$F$-indexed coproducts,” regarding $E$ and $F$ as $F$-indexed categories.

(This is C3.3.3 in the Elephant.)

Proof

As explained at locally connected site, when $C$ is locally connected, the left adjoint ${\Pi }_0:\mathrm{Sh}(C)\to \mathrm{Set}$ is simply obtained by taking colimits over $C^{\mathrm{op}}$. Now by the co-Yoneda lemma, the colimit over any representable presheaf is a singleton (i.e. a terminal object in Set):

But if $C$ has a terminal object, then that terminal object represents the terminal presheaf, which is also the terminal presheaf. Hence under these conditions, ${\Pi }_0$ preserves the terminal object, so $\mathrm{Sh}(C)$ is connected.

Proof

Since the functor ${\mathrm{Topos}}^{\mathrm{op}}\to \mathrm{Cat}$ sending a topos to itself and a geometric morphism to its inverse image functor is 2-fully-faithful (an equivalence on hom-categories), connected morphisms are representably co-fully-faithful in $\mathrm{Topos}$.

Therefore, for 2-categorical orthogonality it suffices to show that in any commutative (up to iso) square

of geometric morphisms in which $p$ is connected and $q$ is etale, there exists a filler $h:C\to B$ such that $hp\cong f$ and $qh\cong g$.

However, if $X\in D$ is such that $B\cong D/X$ (such exists by definition of $q$ being etale), then for any topos $E$ equipped with a geometric morphism $k:E\to D$, lifts of $k$ along $q$ are equivalent to morphisms $*\to k^{*}(X)$ in $C$. In particular, $f$ is determined by a map $*\to f^{*}(q^{*}(X))\cong p^{*}(g^{*}(X))$, and since $*\cong p^{*}(p_{*}(*))$ and $p^{*}$ is fully faithful, this map comes from a map $*\to g^{*}(X)$ in $C$, which in turn determines a geometric morphism $h:C\to B$ which is the desired filler.

Proof

Given $f:E\to S$ locally connected, we can factor it as $E\to S/f_{!}(*)\to S$. The second map is etale by definition, while the first is locally connected (the left adjoint is essentially $f_{!}$ again) and connected since it preserves the terminal object (by construction).

Proof

For every object $X$, we have that $ℰ/X$ sits over $ℰ$ by the etale geometric morphism.

This makes $ℰ/X$ be a locally connected topos.

Notice that the terminal object of $ℰ/X$ is $(X\stackrel{\mathrm{Id}}{\to }X)$. If now $X$ is connected, then

and so the extra left adjoint ${\Pi }_0\circ X_{!}$ preserves the terminal object. By the above proposition this means that $ℰ/X$ is also connected.