Local to global theorem for quasi isometry I am looking for a local - to - global principle in quasi-isometry.
Suppose $Z$ and $X$ are proper, geodesic, hyperbolic metric spaces such there is a local quasi-isometry  $f: Z \to X$. That it for some fixed $R>0$ the map $f$ takes $R$-balls in $Z$ quasi-isometrically onto its image in $X$.
From this is it possible to conclude that $Z$ maps quasi isometrically onto $f(Z) \subset X$?
 A: Here is the correct statement: There exist functions  $R=R(\delta, L, A), L'=L'(\delta, L, A), A'=A'(\delta, L, A)$ such that the following holds:
Theorem. Suppose that $X, Y$ are geodesic  metric spaces, $Y$ is $\delta$-hyperbolic and $f: X\to Y$ is $R$-local $(L,A)$ quasi-isometric embedding (i.e. the restriction of $f$ to every $R$-ball in $X$ is an $(L,A)$ quasi-isometric embedding). Then $f: X\to Y$ is an $(L',A')$ quasi-isometric embedding.
This is a direct consequence of the local-to-global principle for quasi-geodesics in hyperbolic spaces which you can find for instance in
Coornaert, Michel; Delzant, Thomas; Papadopoulos, Athanase, Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov. (Geometry and group theory. The hyperbolic groups of Gromov), Lecture Notes in Mathematics, 1441. Berlin etc.: Springer-Verlag. x, 165 p. (1990). ZBL0727.20018.
On the other hand, without specifying the order of dependence of constants (as in the OP), the answer will be negative:
Let $X$ be the hyperbolic plane in the upper half-plane model $\{(x,y): y>0\}$ and $Y\cong {\mathbb R}$ be the hyperbolic geodesic $\{(0,y): y>0\}$ in $X$. Consider the projection
$$
f(x,y)=(0,y), f: X\to Y.
$$
Then $f$ is $1$-Lipschits and restricts to an $(1,D)$-quasi-isometric embedding on every hyperbolic disk of diameter $D$ in $X$. However, $f: X\to Y$ is clearly not a quasi-isometric embedding.
