  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), that every
Stacey crossed product is an Exel crossed product, and answer some questions
raised by Ionescu and Muhly. While this manuscript is not yet in its final
form(s) several people have already used our results, so we are posting it now.
