I am having trouble following the argument in Massey for the proof of Theorem 8.3, which states that if $f: X \rightarrow Y$ is a homotopy equivalence, then $f_*: \pi(X, x) \rightarrow \pi(Y, f(x))$ is an isomorphism for any $x \in X$.
The proof in the book assumes Theorem 8.2., which states that if $\phi_0, \phi_1: X \rightarrow Y$ are continuous maps which are homotopic, then the image of $\pi(X, x_0)$ under $\phi_{0*}$ and $\phi_{1*}$ are isomorphic.
They use this to claim that $\pi(X, x) \rightarrow \pi(X, gf(x))$ factors through $\pi(Y, f(x))$ via $f_*$ and $g_*$. I do not see why this is true. It seems that theorem 8.2 guarantees that a different diagram commutes.