A Gromov-hyperbolic space $X$ is called *ultracomplete* if any two distinct points $x,y\in\overline{X}$ in the Gromov bordification can be connected by a geodesic segment, ray or line (depending on whether the end points are on the boundary or not). A segment, ray or line is *geodesic* if it is isometric to a closed interval in $\mathbb{R}$.

In $\S$7.5 from Hyperbolic groups, Gromov states without proof that "by a trivial argument", one has the following embedding:

Every geodesic hyperbolic $X$ isometrically embeds into some ultracomplete space $Y$, such that $\mathrm{dist}_Y(y,X)\leq C<\infty$ for a constant $C$ and all $y\in Y$. In particular, $\partial Y=\partial X$. Furthermore, one can choose a $Y$, such that every isometry of $X$ extends to a unique isometry of $Y$.

I understand this statement by two parts: first part is that one can "add" something, not much, to $X$ to make it ultracomplete while preserving the hyperbolicity; the second part is that there is a universality in this extension.

I suppose that the embedding in the first part is "trivially" achieved via a $(1,C)$-quasi-isometry. But I cannot find any reference that gives details. So I came to MO to see if anyone can provide one.