Let $M_X = (X, d_X)$ and $M_Y = (Y, d_Y)$ each be a D1107: Metric space.
The set of Lipschitz constants for $f$ with respect to $M_X$ and $M_Y$ is the D11: Set $$\begin{equation} \left\{ L \in [0, \infty) \mid \forall \, x, y \in X : d_Y ( f(x), f(y)) \leq C d_X(x, y) \right\} \end{equation}$$