Bonjour.
For $i=1,2$ let $X_i$ be a non-empty set and $d_i$ a metric $X_i^2 \to \mathbb{R}$. Suppose $f$ is a locally Lipschitz (*) function $(X_1, d_1) \to (X_2, d_2)$.
Question. Do there exist metrics $\delta_1: X_1^2 \to \mathbb{R}$ and $\delta_2: X_2^2 \to \mathbb{R}$ such that i) $\delta_i$ is (topologically) equivalent to $d_i$ (for $i=1,2$) and ii) $f$ is a Lipschitz function $(X_1, \delta_1) \to (X_2, \delta_2)$?
(*) For each $x \in X_1$, there exists a neighborhood $U_x$ of $x$ in $(X_1, d_1)$ such that the restriction of $f$ to $U_x$ is a Lipschitz function $(U_x, d_1) \to (X_2, d_2)$.