Proposition

Show that the change-of-basepoint homomorphism $\beta_h$ depends only on the homotopy class of $h$.

Solution

Let $h \simeq g$. By the inverse lemma, $\overline{h} \simeq \overline{g}$. The product operation respects homotopy classes, $h \cdot f \cdot \overline{h} \simeq g \cdot f \cdot \overline{g}$. Thus $[h \cdot f \cdot \overline{h}] = [g \cdot f \cdot \overline{g}]$, so $\beta_h(f) = \beta_g(f)$.