Massey Exercise 2.3.1
Let and let and be two path classes from to . We can characterize the conditions under which and give rise to the same isomorphism by observing the following.
Suppose and are the same. Then we have, for all :
Denote , and notice that and . Then we have . Since this holds for all , we have that .
By assuming that , we can reverse the argument above to conclude that .
So we have that the isomorphisms induced by and are identical if and only if .