Theorem 7.9:

Suppose $\gamma_0^\prime, \gamma_0^{\prime \prime}$ are two paths joining $x_1$ and $x_2$ and $h^\prime, h^{\prime \prime}$ are the corresponding group isomorphisms from $\pi_1(X,x_1) \longrightarrow \pi_1(X,x_2)$ given by the previous theorem. Then there exists an inner automorphism

$$\sigma : \pi_1(X,x_2) \longrightarrow \pi_1(X,x_2)$$

such that $h^\prime= \sigma \circ h^{\prime \prime}.$ In fact $\sigma$ is the inner automorphism determined by $[\gamma_0^\prime][\gamma_0^{\prime \prime}]^{-1}.$

If $\pi_1(X, x_0)$ is abelian then $\pi_1(X, x_0)$ and $\pi_1(X,x_1)$ are naturally isomorphic. That is the isomorphism $h_{[\sigma]}$ is canonical in this case.