We recently defined an invariant of contact manifolds with convex boundary in
Kronheimer and Mrowka's sutured monopole Floer homology theory. Here, we prove
that there is an isomorphism between sutured monopole Floer homology and
sutured Heegaard Floer homology which identifies our invariant with the contact
class defined by Honda, Kazez and Mati\'c in the latter theory. One consequence
is that the Legendrian invariants in knot Floer homology behave functorially
with respect to Lagrangian concordance. In particular, these invariants provide
computable and effective obstructions to the existence of such concordances.
Our work also provides the first proof which does not rely on the relative
Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and
Mati\'c's contact class is a well-defined invariant of contact manifolds.