  The contact invariant is an element in the monopole Floer homology groups of
an oriented closed three manifold canonically associated to a given contact
structure. A non-vanishing contact invariant implies that the original contact
structure is tight, so understanding its behavior under symplectic cobordisms
is of interest if one wants to further exploit this property. By extending the
gluing argument of Mrowka and Rollin to the case of a manifold with a
cylindrical end, we will show that the contact invariant behaves naturally
under a strong symplectic cobordism. As quick applications of the naturality
property, we give alternative proofs for the vanishing of the contact invariant
in the case of an overtwisted contact structure, its non-vanishing in the case
of strongly fillable contact structures and its vanishing in the reduced part
of the monopole Floer homology group in the case of a planar contact structure.
We also prove that a strong filling of a contact manifold which is an L-space
must be negative definite.
