Hyperbolicity is transported by a quasi isometry

Fix any geodesic triangle in $X$ with vertices $a,b,c$, and any point $p$ on its specified side $[a,b]$. Compose the isometric parametrization of each of the three sides with $f$. The given inequalities make each composition a $(\lambda ,\epsilon )$-quasi-geodesic on a nonempty compact interval, even if $f$ is discontinuous. Choose three target endpoint geodesics $G_{ab},G_{bc},G_{ca}$. F1 applies separately to every one of them.

Fix $h>0$. The first Hausdorff inclusion gives $p^{\mathrm{\prime }}\in G_{ab}$ with $d_Y(f(p),p^{\mathrm{\prime }})<M+h$. Target slimness gives $q^{\mathrm{\prime }}\in G_{bc}\cup G_{ca}$ with $d_Y(p^{\mathrm{\prime }},q^{\mathrm{\prime }})<\delta +h$. The reverse Hausdorff inclusion on whichever side contains $q^{\mathrm{\prime }}$ gives $q\in [b,c]\cup [c,a]$ with $d_Y(q^{\mathrm{\prime }},f(q))<M+h$. Hence $d_Y(f(p),f(q))<2M+\delta +3h$, and the lower embedding inequality gives $d_X(p,q)<\lambda (2M+\delta +\epsilon +3h)\mathrm{.}$ Only finitely many approximate witnesses are used; there is no assumption that the image of $f$ or of a side has closest points.

Taking the infimum over the other two source sides and then letting $h$ decrease to zero proves their distance from $p$ is at most $\lambda (2M+\delta +\epsilon )$. The chosen source triangle, side and point were arbitrary, proving the exact stated slimness constant. Repeated vertices and zero parameters are included: a side can have a one-point interval, and the argument divides only by the positive $\lambda$.

Finally let $f$ be a quasi-isometry in F3's convention, with supplied coarse inverse $h$. If $h$ has coarse Lipschitz constants $A,B$ and $d_X(hf(x),x)\le D$, then $d_X(x,x^{\mathrm{\prime }})\le 2D+A{d}_Y(fx,f{x}^{\mathrm{\prime }})+B\mathrm{.}$ Replace $A$ by $\max \{1,A\}$, combine this lower bound for $f$ with its coarse Lipschitz upper bound, and enlarge the two constants to obtain some $(\lambda ,\epsilon )$ embedding inequalities. The other composite bound gives attained coarse density. F2 therefore supplies a controlled inverse $g$ which is a quasi-isometric embedding. If $Y$ is hyperbolic, step 3.1 applied to $f$ proves $X$ hyperbolic; if $X$ is hyperbolic, apply the same result to $g$ to prove $Y$ hyperbolic. If the spaces are empty, F3 forces both empty and there are no triangles to check. This proves invariance with the stated AC assumption.