Let (X, S), (Y, T) be topological dynamical systems and \pi : X \rightarrow Y a factor map. A function F \in C(X) is a compensation function if P (F + \phi \circ \pi) = P (\phi) for all \phi \in C(Y). We present an example of a factor map \pi : X \rightarrow Y between subshifts of finite type X, Y that does not have a saturated compensation function and an example of a non-Markovian factor map with a saturated compensation function. Also we provide a necessary and sufficient condition for a certain type of factor map to have a saturated compensation function.