Let $n\in\mathbb{N}$ a fixed natural number and $I_\varphi\subset\mathbb{R}$ a fixed compact interval. Consider the space of all $\cal{C}^1$ functions $\varphi:I_\varphi\to \mathbb{R}^n$. Define an equivalence relation as $\varphi \cal{R} \phi$ iff there exists a function $\lambda:I_\varphi\to \phi$ that is bijective, $\lambda \in \cal{C}^1(I_\varphi), \lambda^{-1} \in \cal{C}^1(I_\phi)$ and $\varphi = \phi \circ \lambda$. This application $\lambda$ is called change of parameter.
Is the change of parameter between $\phi$ and $\varphi$ unique? The above are the definitions I was given, but I suppose we are not considering 'degenerated curves', that is, Im $\varphi$ is not a single point.
My guess is that there is uniqueness, but I couldn't conclude. My attemp: $$ \varphi \circ \lambda_1 = \varphi \circ \lambda_2 \implies \varphi \circ \lambda_1 \circ \lambda_2^{-1} = \varphi $$
But how does this imply that $\lambda_1 \circ \lambda_2^{-1} = Id$?