Corollary 11.4:

Suppose that $F$ is a homotopy between maps $f, g : X \longrightarrow Y$ and for a point $x_0 \in X$, $f(x_0) = g(x_0) = y_0$. If $\pi_1(Y, y_0)$ is abelian then the group homomorphisms $f_*$ and $g_*$ are equal.

If we drop the hypothesis $f(x_0) = g(x_0)$ in theorem 11.3 the proof still goes through but since $\sigma$ is no longer a loop we merely get that the induced maps $f_*$ and $g_*$ differ by a composition through the isomorphism $h_{[\sigma]}$ encountered in theorem (7.8). We record the result as a theorem and the reader may rework the proof of theorem 11.3 to fit it in the present context.