Proof:

Let $\sigma$ be a path joining $x_1$    and $x_2$. Observe that if $\gamma$ is a loop at the $x_1$ then $\sigma \ast \gamma \ast \sigma^{-1}$ is a loop at $x_2$ thereby enabling us to define a map

$$h_{[\sigma]} :\; \pi_1(X,x_1) \longrightarrow \pi_1(X,x_2)$$

$$[\gamma]\;\mapsto [\sigma \ast \gamma \ast \sigma^{-1}].$$

in Corollary (7.4) shows that the function is well defined and lemma (7.5) shows that it is a group homomorphism. Let $\Gamma$ be a loop at $x_2.$ Then $\sigma^{-1} \ast \Gamma \ast \sigma$ is a loop at $x_1$ and $h_{[\sigma]}([\sigma^{-1} \ast \Gamma \ast \sigma])=[\Gamma]$ showing that $h_{[\sigma]}$ is surjective. The map

$$h_{[\sigma^{-1}]}:[\Gamma] \longrightarrow [\sigma^{-1} \ast \Gamma \ast \sigma]$$

is the inverse of $h_{[\sigma]}$. $\square$