Hyperbolicity is a quasi-isometry invariant of geodesic spaces

Assume $X$ is hyperbolic and let $f\text{\hspace{0.17em}⁣}:X\to Y$ be the given quasi-isometry. By [A1], choose a quasi-inverse $g\text{\hspace{0.17em}⁣}:Y\to X$. For any geodesic triangle in $Y$, the images of its sides under $g$ are uniform quasi-geodesics in $X$.

Since $X$ is hyperbolic, [L1] shows that each of those quasi-geodesic sides stays within a bounded distance of a genuine geodesic triangle in $X$. Applying $f$ back to that comparison triangle produces a bounded-neighborhood comparison in $Y$, because $f$ is coarsely Lipschitz and $f\circ g$ stays uniformly close to the identity on $Y$.

Therefore geodesic triangles in $Y$ are uniformly slim, so $Y$ is hyperbolic. Reversing the roles of $X$ and $Y$ gives the converse. Hence hyperbolicity is a quasi-isometry invariant of geodesic spaces.