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Lipshitz, Ozsv\'ath and Thurston defined a bordered Heegaard Floer invariantCFDA for 3-manifolds with two boundary components, including mapping cylindersfor surface diffeomorphisms. We define a related invariant for certain4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f)to CFDA(g) to each such cobordism between two mapping cylinders f and g. Likethe Ozsv\'ath-Szab\'o invariants of cobordisms between closed 3-manifolds, thismorphism arises from counting holomorphic triangles on Heegaard triples. Wedemonstrate that the homotopy class of the morphism F only depends on thesymplectic structure of the cobordism in question.

Bordered Floer Homology and Lefschetz fibrations with corners