Show that the change-of-basepoint homomorphism $\beta_h$ depends only on the homotopy class of $h$.
Definition- A change-of-basepoint map $\beta_h : \pi_1(X,x_1)\rightarrow \pi_1(X,x_0)$
by $\beta_h[f]= [h \cdot f \cdot \overline{h}].$ This is well-deﬁned since if $f_t$ is a homotopy of loops based at $x_1$ then $h \cdot f_t \cdot \overline{h}$ is a homotopy of loops based at $x_0.$