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Let C be a semidualizing module over a commutative noetherian ring R. We exhibit an isomorphism between the bifunctors defined via C-flat and C-projective resolutions. We show how the vanishing of these functors characterizes the finiteness of , and use this to give a relation between the of a module and of a pure submodule. On the other hand, we show that other isomorphisms force C to be trivial.