I have several questions about geometric morphisms of topoi. It was recommended that I move my question here from Math.Stackexchange, since it would be good to get an expert on topos theory to answer.
1) A geometric morphism $g \dashv f : C \rightarrow D$, with $f : C \rightarrow D$ is called strongly connected if $g$ has a further left adjoint $h : C \rightarrow D$ which preserves finite products. This is equivalently an essential geometric morphism in which the left-most adjoint preserves finite products. Do people ever consider geometric morphisms $g \dashv f : C \rightarrow D$, with $f : C \rightarrow D$, where $f : C \rightarrow D$ has a right adjoint $h : C \rightarrow D$ which preserves finite coproducts? This seems like the natural dual of strongly connected, but does it have any significance in topos theory?
2) Is there the opposite notion of a local geometric morphism? That would seem to be a geometric morphism $g \dashv f : C \rightarrow D$ which has a further left adjoint $h \dashv g$, $h : C \rightarrow D$ such that $h$ is fully faithful.
3) I am curious about the dual to essential morphisms as well.
I expect that local geometric morphisms are more significant than their opposite cousins, and the same for strongly connected morphisms, since topoi are not self dual. Still, an explanation of the "asymmetrical significance" would help me to understand why we consider one and not the other.