  In this article we extend the construction of the Floer fundamental group to
the monotone Lagrangian setting and use it to study the fundamental group of a
Lagrangian cobordism $W\subset (\mathbb{C}\times M, \omega_{st}\oplus\omega)$
between two Lagrangian submanifolds $L, L'\subset ( M, \omega)$. We show that
under natural conditions the inclusions $L,L'\hookrightarrow W$ induce
surjective maps $\pi_{1}(L)\twoheadrightarrow\pi_{1}(W)$,
$\pi_{1}(L')\twoheadrightarrow\pi_{1}(W)$ and when the previous maps are
injective then $W$ is an h-cobordism.

Symplectic Geometry