Topoi of sheaves for (finitary) coherent topologies on coherent categories are called coherent toposes. (The terminology is slightly confusing, though, because every topos is a coherent category.) Note that every topos is equivalent to a topos of sheaves for the infinitary coherent topology on an infinitary-coherent site, namely itself.
If is a pretopos, then its self-indexing is a stack for its coherent topology. Exactness and extensivity are stronger than necessary, however; a pair of necessary and sufficient conditions for this to hold are that
If is a kernel pair, then for any , the pullback is also a kernel pair (this is equivalent to the codomain fibration being a stack for the regular topology).
If are a pair of subobjects, then for any and and any isomorphism , the pushout