Let $f$ be a loop based at $x_0$ and $c_{x_0}$ be the constant loop. Construct an explicit homotopy between $f$ and $f\cdot c_{x_0}$.
I have seen that $$F(s,t)=\begin{cases}f\big(\frac{s}{1-\frac{t}{2}} \big)\ \ \text{if}\ 0\leq s\leq 1-\frac{t}{2} \\ x_0\ \ \ \ \ \ \ \ \ \ \ \text{if}\ 1-\frac{t}{2}\leq s\leq 1\end{cases}$$ is a homotopy between $f$ and $f\cdot c_{x_0}$. But how to get that homotopy? Is there any explicit way to construct that?