Homotopy between loop and composite with constant loop

Statement

Existential version

Suppose $x_{0}$ is a point in a topological space $X$ and $f$ is a loop based at $x_{0}$, i.e., $f$ is a continuous map from $[0,1]$ to $X$ such that $f(0)=f(1)=x_{0}$. Suppose $e$ is the constant loop based at $x_{0}$, i.e., the loop that stays at $x_{0}$ throughout.

Denote by $*$ the composition of loops by concatenation. Then, $f$ is homotopic to the loops $e*f$ and $f*e$.

Graphical version

Here is a pictorial description of the homotopy between $f$ and $e*f$:

Here is a pictorial description of the homotopy between $f$ and $f*e$: