Let $X,Y$ be metric space with $X$ $C$-doubling; i.e: each $B_X(x,r)$ can be covered by at-most $C$ balls of radius $B_X(x,r/2)$. Let $f:X\rightarrow Y$ be a Lipschitz map, then clearly $f(X)$ is also doubling, but how does its doubling constant (previously $C$) change under $f$?