Theorem 11.3:

Suppose that $F$ is a homotopy between maps $f, g : X \longrightarrow Y$ and for a point $x_0 \in X$, $f(x_0) = g(x_0) = y_0$. Then the group homomorphisms $f_*$ and $g_*$ are conjugate by the inner-automorphism generated by the loop

$$\sigma: t\mapsto F(x_0, t) \eqno(11.3)$$