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This is the third of five papers that construct an isomorphism between theSeiberg-Witten Floer homology and the Heegaard Floer homology of a givencompact, oriented 3-manifold. The isomorphism is given as a composition ofthree isomorphisms; the first of these relates a version of embedded contacthomology on an an auxillary manifold to the Heegaard Floer homology on theoriginal. This paper describes the relationship between the differential on theembedded contact homology chain complex and the differential on the HeegaardFloer chain complex. The paper also describes the relationship between thevarious canonical endomorphisms that act on the homology groups of these twocomplexes.

HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence