  A marked free monoid morphism is a morphism for which the image of each
generator starts with a different letter, and immersions are the analogous maps
in free groups. We show that the (simultaneous) PCP is decidable for immersions
of free groups, and provide an algorithm to compute bases for the sets, called
equalisers, on which the immersions take the same values. We also answer a
question of Stallings about the rank of the equaliser.
  Analogous results are proven for marked morphisms of free monoids.
