  There are many different crossed products by an endomorphism of a C*-algebra,
and constructions by Exel and Stacey have proved particularly useful. Here we
show that every Exel crossed product is isomorphic to a Stacey crossed product,
though by a different endomorphism of a different C*-algebra. We apply this
result to a variety of Exel systems, including those associated to shifts on
the path spaces of directed graphs.
