A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative 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.