  We study the maps induced on link Floer homology by elementary decorated link
cobordisms. We compute these for births, deaths, stabilizations, and
destabilizations, and show that saddle cobordisms can be computed in terms of
maps in a decorated skein exact triangle that extends the oriented skein exact
triangle in knot Floer homology. In particular, we completely determine the
Alexander and Maslov grading shifts.
  As a corollary, we compute the maps induced by elementary cobordisms between
unlinks. We show that these give rise to a $(1+1)$-dimensional TQFT that
coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of
resolutions of a marked link diagram, it gives the complex defining the reduced
Khovanov homology of the knot. Finally, we define a spectral sequence from
(reduced) Khovanov homology using these cobordism maps, and we prove that it is
an invariant of the (marked) link.
