Theorem 11.2:

Suppose that $f$ and $g$ are homotopic maps of pairs $(X, x_0)$ and $(Y, y_0)$ then the induced group homomorphisms $f_*$ and $g_*$ from $\pi_1(X, x_0)$ to $\pi_1(Y, y_0)$ are equal.

Now suppose that $f$ and $g$ are homotopic maps from $X$ to $Y$ such that for a base point $x_0 \in X$, $f(x_0) = g(x_0) = y_0$ say, but the intermediate maps do not respect these base points. Then it is not necessary that $f_* = g_*$ as maps from $\pi_1(X, x_0)$ to $\pi_1(Y, y_0)$. The following theorem addresses this issue.