Lemma 9.1:

Suppose that $(X, x_0), (Y, y_0)$ and $(Z, z_0)$ are pointed topological spaces. Let $f: (X, x_0) \longrightarrow (Y, y_0)$ and $g:(Y, y_0) \longrightarrow (Z, z_0)$ be continuous maps of pairs, that is continuous maps satisfying $f(x_0) = y_0;\; g(y_0) = z_0$, then the induced homomorphisms on the respective fundamental groups satisfies

$$(g \circ f)_\ast = g_{\ast} \circ f_{\ast}.$$

If id$_{x} : X \longrightarrow X$ is the identity map then $($id$_{x})_{\ast}=$   id$_{\pi_1(X,x_0)}$. That is to say, the identity map on $X$ induces the identity homomorphism on $\pi_1(X, x_0)$.