  We construct a product on the Floer complex associated to a pair of
Lagrangian cobordisms. More precisely, given three exact transverse Lagrangian
cobordisms in the symplectization of a contact manifold, we define a map
$\mathfrak{m}_2$ by a count of rigid pseudo-holomorphic disks with boundary on
the cobordisms and having punctures asymptotic to intersection points and Reeb
chords of the negative Legendrian ends of the cobordisms. More generally, to a
$(d+1)$-tuple of exact transverse Lagrangian cobordisms we associate a map
$\mathfrak{m}_d$ such that the family $(\mathfrak{m}_d)_{d\geq1}$ are
$A_\infty$-maps. Finally, we extend the Ekholm-Seidel isomorphism to an
$A_\infty$-morphism, giving in particular that it is a ring isomorphism.
