Let $f,g: [0,1]^n=I^n \rightarrow X$ be contiuous maps s.t. $f(\partial I^n)=g(\partial I^n)=x_1$. If $\gamma:I \rightarrow X$ is a path joining $x_0$ and $x_1$ $\big(\gamma(0)=x_0,\gamma(1)=x_1\big)$.
Now we define the map $f_{\gamma}:(I^n, \partial I^n)\rightarrow (X,x_0)$ as:
$f_{\gamma}(t_1,\cdots,t_n)=f(2t_1-\frac{1}{2},\cdots,2t_n-\frac{1}{2})$, if $|t|:=max\{|2t_1-1|,...,|2t_n-1|\}\leq\frac{1}{2}$
$f_{\gamma}(t_1,\cdots,t_n)=\gamma(2-2|t|)$, if $|t|\geq\frac{1}{2}$
I'm trying to prove that $f_{\gamma}*g_{\gamma} \simeq (f *g)_{\gamma}$ $(rel(\partial I^n))$, $f_{\gamma \circ \eta}\simeq (f_{\eta})_{\gamma}$ $(rel(\partial I^n))$ and $f_{c_{x_1}}\simeq f$ $(rel(\partial I^n))$, where $\eta$ is a path s.t. $\gamma(1)=\eta(0), {c_{x_1}}$ is the path constantly $x_1$ and $*$ denotes the usual path-composition.
What I don't know how to do is giving explicit formulas for homotopies, so I'd really appreciate any help with it. Thanks