  We prove that the Seidel morphism of $(M \times M', \omega \oplus \omega')$
is naturally related to the Seidel morphisms of $(M,\omega)$ and
$(M',\omega')$, when these manifolds are monotone. We deduce that any homotopy
class of loops of Hamiltonian diffeomorphisms of one component, with
non-trivial image via Seidel's morphism, leads to an injection of the
fundamental group of the group of Hamiltonian diffeomorphisms of the other
component into the fundamental group of the group of Hamiltonian
diffeomorphisms of the product. This result was inspired by and extends results
obtained by Pedroza [P08].
