Let $M_X = (X, d_X)$ and $M_Y = (Y, d_Y)$ each be a D1107: Metric space.
A D18: Map $f : X \to Y$ is a bilipschitz map from $M_X$ to $M_Y$ if and only if $$\begin{equation} \exists \, C \in [1, \infty) : \forall \, x, y \in X : \frac{1}{C} d_X(x, y) \leq d_Y ( f(x), f(y) ) \leq C d_X(x, y) \end{equation}$$