The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map \Phi^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded contact homology chain complex for a contact form supported by (S,h). The chain map \Phi^+ commutes up to homotopy with the U-maps defined on both sides and reduces to the quasi-isomorphism \Phi from "The equivalence of Heegaard Floer homology and embedded contact homology I, II" on subcomplexes defining the hat versions. Algebraic considerations then imply that the map \Phi^+ is a quasi-isomorphism.