Given three metric spaces $(X,d_X),(Y,d_Y),(Z,d_Z)$ and an embedding function $g: X \rightarrow Z$, I am interested in finding more information about functions $f: X \rightarrow Y$ in which there exists $M>0$ such that $\forall x,y \in X$ satisfy $$d_Y(f(x),f(y)) \leq M d_Z(g(x),g(y))$$
In other words, a Lipschitz continuous function in the embedding space. I am having difficulties finding any references to functions like this. I have seen a reference to something like this in a paper (http://arxiv.org/pdf/2203.13270) but, to my knowledge, they invented a new term and there was no prior literature.