Given a finite state transducer defining a rational relation over infinite words, it is known to be decidable whether or not the relation is a function, i.e. whether each infinite input word is related to at most one infinite output word. This is detailed in a paper by Gire: Two Decidability Problems for Infinite Words. Unfortunately, I cannot find the full text of the paper anywhere. The basic idea seems to be to form the composition of the transducer with its inverse $T \circ T^{-1}$ and check if the resulting transducer is a restriction of the identity function. Note that the inverse of a transducer $T$ is $T$ with input and output word swapped for each transition.
I am looking for details of the decision procedure. Do you have any references or a short description of the algorithm?