A Grothendieck 2-topos is a 2-category that is equivalent to the 2-category of 2-sheaves (aka stacks) on some 2-site. More generally, a Grothendieck -topos is an -category that is equivalent to the -category of -sheaves on some -site (for ); see truncated 2-topos.
The most familiar Grothendieck 2-toposes are those of 2-sheaves on a 1-site. Even 2-presheaves on a 2-category that is not a 1-category can exhibit unexpected behavior; for instance they do not in general have cores or opposites. (2-presheaves on a (2,1)-category do have cores and opposites, though.)
The 2-Giraud theorem implies that if is a Grothendieck 2-topos, so is . In the case of 2-presheaves, we can identify this as:
since we have . In other words, the “2-topos of presheaves” functor from 2-categories to 2-toposes preserves the 2-cell-dual involution. Another way to say this is that only for 2-sheaves on a (2,1)-category is “oppositization” an endofunctor; in other cases “passage to opposites” means passage to a different 2-topos.