Show that the change-of-basepoint homomorphism $\displaystyle \beta_h$ depends only on the homotopy class of $\displaystyle h$.

Definition-A change-of-basepoint map $\displaystyle \beta_h : \pi_1(X,x_1)\rightarrow \pi_1(X,x_0)$

by $\displaystyle \beta_h[f]= [h \cdot f \cdot \overline{h}].$ This is well-deﬁned since if $\displaystyle f_t$ is a homotopy of loops based at $\displaystyle x_1$ then $\displaystyle h \cdot f_t \cdot \overline{h}$ is a homotopy of loops based at $\displaystyle x_0.$

Update: Solved it.