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Categorical aspects of bivariant K-theory 17Given correspondences E from A to B and F from B to C , their composition isthe correspondence from A to C with underlying Hilbert module E x˝BK F and mapA K ! K.E/ ! K.E x˝BK F /, where the last map sends T 7! T ˝ 1; this yieldscompact operators because B K maps to K.F /. See [34] for the definition of the relevantcompleted tensor product of Hilbert modules.Up to isomorphism, the composition of correspondences is associative and the identitymaps A ! A D K.A/ act as unit elements. Hence we get a category Corr G whosemorphisms are the isomorphism classes of correspondences. It may have advantagesto treat Corr G as a 2-category.Any -homomorphism ' W A ! B yields a correspondence f W A ! K.E/ from Ato B, so that we get a canonical functor \W G-C alg ! Corr G . We let E be the rightideal '.A K / B K in B K , viewed as a Hilbert B-module. Then f.a/ b WD '.a/ brestricts to a compact operator f.a/ on E and f W A ! K.E/ is essential. It can bechecked that this construction is functorial.In the following proposition, we require that the category of G-C -algebras S beclosed under Morita–Rieffel equivalence and consist of -unital G-C -algebras. Welet Corr S be the full subcategory of Corr G with object class S.Proposition 39. The functor \W S ! Corr S is the universal C -stable functor on S;that is, it is C -stable, and any other such functor factors uniquely through \.Proof. First we sketch the proof in the non-equivariant case.We verify that \ is C -stable. The Morita–Rieffel equivalence between K.`2N/ ˝A Š K `2.N;A/ and A is implemented by the Hilbert module `2.N;A/, whichyields a correspondence id;`2.N;A/ from K.`2N/ ˝ A to A; this is inverse to thecorrespondence induced by a corner embedding A ! K.`2N/ ˝ A.A Hilbert B-module E with an essential -homomorphism A ! K.E/ is countablygenerated because A is assumed -unital. Kasparov’s Stabilisation Theorem yields anisometric embedding E ! `2.N;B/. Hence we get -homomorphismsA ! K.`2N/˝ B B:This diagram induces a map F .A/ ! F.K.`2N/ ˝ B/ Š F.B/ for any C -stablefunctor F . Now we should check that this well-defines a functor xF W Corr S ! C withxF ı \ D F , and that this yields the only such functor. We omit these computations.The generalisation to the equivariant case uses the crucial property of the left regularrepresentation that L 2 .G/ ˝ H Š L 2 .G N/ for any countably infinite-dimensionalG-Hilbert space H. Since we replace A and B by A K and B K in the definition ofcorrespondence right away, we can use this to repair a possible lack of G-equivariance;similar ideas appear in [36].Example 40. Let u be a G-invariant multiplier of B with u u D 1; such u arealso called isometries. Then b 7! ubu defines a -homomorphism B ! B. Theresulting correspondence B Ü B is isomorphic as a correspondence to the identity