Basic notions of coalgebras and modal logic are reviewed.

A categorical description of coalgebraic semantics of modal logics is developed.

This description turns out to dualise algebraic semantics of equational logic as given by Banaschewski and Herrlich in the paper "Subcategories defined by implications" (1976). This gives a precise meaning to the slogan that "Modal Logic is Dual to Equational Logic".

Theorems characterising the expressive power of infinitary modal logics on Kripke-frames can now be obtained by dualsing results of Banaschewski and Herrlich.