We show for a given metric space $(X,d)$ of asymptotic dimension $n$ that there exists a coarsely and topologically equivalent hyperbolic metric $d'$ of the form $d' = f \circ d$ such that $(X,d')$ is of asymptotic Assouad-Nagata dimension $n$. As a corollary we construct examples of spaces realising strict inequality in the logarithmic law for AN-asdim of a Cartesian product. One of them may be viewed as a counterexample to a specific kind of a Morita-type theorem for AN-asdim.

A superfluous step in the proof of 1.4 removed, remarks about earlier results updated, a reference to the BSc thesis added, minor corrections