Recall: A frame for the basic modal language is a pair with a non-empty set and a binary relation on .
At their ‘crudest’ the objects studied are just sets and binary relations on them. In the basic ‘many worlds’ interpretation, a world satisfies a modal formula if all worlds accessible from (that is satisfying ) satisfy .
Now convert into a mapping , where is the set of worlds accessible from . It is easy to rephrase the notion of ‘satisfies’ in terms of , but is a coalgebra for the power set endofunctor on , so Kripke frames provide a simple example of a coalgebraic modal for modal formulae. There are many other types of coalgebras for endofunctors and many lead to modal logics.