In homotopy theory,suppose $\gamma$ is a loop in $(X, x_0)$, and $f:(I^n,\partial I^n) \to (X, x_0)$ is a map. where $\gamma f$ is defined in the following way in Hatcher's book:
This induced the so-called the action of $\pi_1$ on $\pi_n$. We show that it is well defined, independent of the choice of element in homotopy class.
(1) If $F_t:f_1\simeq f_2:(I^n,\partial I^n) \to (X, x_0)$, then $\gamma F_t$ will give the homotopy: $\gamma f_1\simeq \gamma f_2$.
(2) Now suppose that $G_t:\gamma_1\simeq \gamma_2$ is a homotopy between two loops on $(X,x_0)$, then $G_t f :\gamma_1 f \simeq \gamma_2 f$.
Am I right in (1) and (2)?