  Hopsets and spanners are fundamental graph structures, playing a key role in
shortest path computation, distributed communication, and more. A (near-exact)
hopset for a given graph $G$ is a (small) subset of weighted edges $H$ that
when added to the graph $G$ reduces the number of hops (edges) of near-exact
shortest paths. Spanners and distance preservers, on the other hand, ask for
removing many edges from the graph while approximately preserving shortest path
distances.
  We provide a general reduction scheme from graph hopsets to the known metric
compression schemes of spanners, emulators and distance preservers.
Consequently, we get new and improved upper bound constructions for the latter,
as well as, new lower bound results for hopsets. Our work makes a significant
progress on the tantalizing open problem concerning the formal connection
between hopsets and spanners, e.g., as posed by Elkin and Neiman [Bull. EATCS
2020].
