We knew that each functor would preserve weak pullbacks if every congruence is a difunctional bisimulation. In chapter 3 we found equivalent statements for weak kernel preservation and the preimage preservation. Additionally, we defined a functor modification, which we called preimage correction. The resulted functor preserves preimages. The idea was inspired by the transformation, which result is a sound functor. The resulted functor has the advantage, that its subfunctors are precisely the subfunctors which preserves preimages. In chapter 4 we showed that the monotonic separable boxes provide a correct and complete modal logic. Interestingly the preimage correction of the general neighborhoodfunctor provides a functor, that preserves weak pullbacks.
