Given a Morse-Smale pair on a manifold M, it is possible to entirely recover its fundamental group in a combinatorial manner. We call this construction the Morse fundamental group. Motivated by a similar construction of a « Floer fundamental group » by Barraud, and by the many uses of continuation maps in symplectic topology, I will explain in this talk how continuation maps give us functoriality and invariance of the Morse fundamental group, and what the differences are with the usual homological setup.