Theorem 11.5:

Suppose that $F$ is a homotopy between maps $f, g : X \longrightarrow Y$ then for $x_0 \in X$, the induced maps $f_*: \pi_1(X, x_0)\longrightarrow \pi_1(Y, f(x_0))$ and $g_*: \pi_1(X, x_0)\longrightarrow \pi_1(Y, g(x_0))$ satisfy the relation

$$h_{[\sigma]}\circ f_* = g_* \eqno(11.4)$$

where $h_{[\sigma]}$ is the isomorphism

$$h_{[\sigma]}: [\gamma] \mapsto [\sigma*\gamma*\sigma^{-1}] \eqno(11.5)$$

we have encountered earlier with $\sigma$ being the curve $F(x_0, t)$ joining $f(x_0)$ and $g(x_0)$.