The Covering Homotopy Property: If is a homotopy such that F(0,0) = 1, then there is a lifting of F to a unique covering homotopy for which .
You have loops f and g in with a base point 1 having the same degree. This implies that the covering paths beginning at common initial point 0 have common terminal point . The following homotopy demonstrates the equivalence of .
By the Covering Homotopy Property, the homotopy pH is a path homotopy between f and g (p is defined in your question).
You need to make sure the following cases if pH is indeed a path homotopy between f and g.
pH(0,s), pH(1,s), pH(t,0), and pH(t,1).