  A correspondence functor is a functor from the category of finite sets and
correspondences to the category of k-modules, where k is a commu-tative ring. A
main tool for this study is the construction of a correspondence functor
associated to any finite lattice T. We prove for instance that this functor is
projective if and only if the lattice T is distributive. Moreover, it has
quotients which play a crucial role in the analysis of simple functors. The
special case of total orders yields some more specific and complete results.
