  This is the third of five papers that construct an isomorphism between the
Seiberg-Witten Floer homology and the Heegaard Floer homology of a given
compact, oriented 3-manifold. The isomorphism is given as a composition of
three isomorphisms; the first of these relates a version of embedded contact
homology on an an auxillary manifold to the Heegaard Floer homology on the
original. This paper describes the relationship between the differential on the
embedded contact homology chain complex and the differential on the Heegaard
Floer chain complex. The paper also describes the relationship between the
various canonical endomorphisms that act on the homology groups of these two
complexes.
