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    • CommentRowNumber1.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJul 7th 2021
    • (edited Apr 11th 2023)

    [deleted]

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2021

    There is a 2-variable adjunction, expressed as usual in terms of hom sets:

    Hom A,C(M⊠ BN,S)≅Hom A,B(M,Hom C R(N,S))≅Hom B,C(N,Hom A L(M,S)) Hom_{A,C}(M \boxtimes_B N, S) \cong Hom_{A,B}(M, \mathbf{Hom}_C^R(N,S)) \cong Hom_{B,C}(N, \mathbf{Hom}_A^L(M,S))

    for MM an (A,B)(A,B)-bimodule, NN a (B,C)(B,C)-bimodule, and SS an (A,C)(A,C)-bimodule. Your isomorphisms can be recovered from this by the Yoneda lemma together with associativity of ⊠\boxtimes:

    Hom D,A(T,Hom C R(M⊠ BN,P)) ≅Hom D,C(T⊠ A(M⊠ BN),P) ≅Hom D,C((T⊠ AM)⊠ BN,P) ≅Hom D,B(T⊠ AM,Hom C R(N,P)) ≅Hom D,A(T,Hom B R(M,Hom C R(N,P))) \begin{aligned} Hom_{D,A}(T, \mathbf{Hom}_C^R(M\boxtimes_B N, P)) &\cong Hom_{D,C}(T\boxtimes_A (M \boxtimes_B N), P)\\ &\cong Hom_{D,C}((T\boxtimes_A M) \boxtimes_B N, P) \\ &\cong Hom_{D,B}(T\boxtimes_A M, \mathbf{Hom}_C^R(N,P)) \\ &\cong Hom_{D,A}(T, \mathbf{Hom}_B^R(M, \mathbf{Hom}_C^R(N,P))) \end{aligned}
    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2021

    The same proof works in any closed bicategory.

    • CommentRowNumber4.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeJul 7th 2021
    • (edited Apr 11th 2023)

    [deleted]

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