Hyperconnected geometric morphism
= Hyperconnected geometric morphism
A <geometric morphism> $h$ is hyperconnected when its <inverse image functor of a geometric morphism> is <full and faithful> and its essential image is closed under <subobjects>. It then identifies all subobjects of $h^*X$ with subobjects of $X$, and its image is closed under quotients as well, by descending the kernel-pair relation.