  Given two correspondences X and Y, we show that (under mild hypotheses) the
Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain
subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the
Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a
natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We
explore implications for graph algebras, crossed products by the integers, and
crossed products by completely positive maps. We also give a new proof of a
result of Kaliszewski and Quigg related to coactions on correspondences.
