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There is a 2-variable adjunction, expressed as usual in terms of hom sets:
HomA,C(M⊠BN,S)≅HomA,B(M,HomRC(N,S))≅HomB,C(N,HomLA(M,S))for M an (A,B)-bimodule, N a (B,C)-bimodule, and S an (A,C)-bimodule. Your isomorphisms can be recovered from this by the Yoneda lemma together with associativity of ⊠:
HomD,A(T,HomRC(M⊠BN,P))≅HomD,C(T⊠A(M⊠BN),P)≅HomD,C((T⊠AM)⊠BN,P)≅HomD,B(T⊠AM,HomRC(N,P))≅HomD,A(T,HomRB(M,HomRC(N,P)))The same proof works in any closed bicategory.
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