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A right-closed bicategory [1] is a bicategory that has all right extensions (i.e. right adjoints to precomposition with a fixed 1-cell). A one-object right-closed bicategory is therefore a right-closed monoidal category. (There are dual notions involving left extensions, and left and right lifts, corresponding similarly to left-closed and left- and right-coclosed monoidal categories.)

Is there a coherence theorem in the literature for right-closed bicategories, or can it be deduced from an existing result? I expect something like the following to be true:

Every right-closed bicategory is biequivalent to a (strictly) right-closed 2-category.

Here "strictly right-closed" would mean something like "composition of right extensions is strictly associative and unital".

[1]: Definition 16.3.1, May–Sigurdsson's Parametrized Homotopy Theory

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    $\begingroup$ FWIW, note that the notion of closed bicategory (or closed monoidal category) is not one for which "all diagrams commute". A counterexample for right-closed monoidal categories can be found in section 5 of Lambek's paper Deductive systems and categories I. However, this example involves noninvertible maps, so it's not necessarily an obstacle to strictifying the associativity/unit isomorphisms for the right extensions as you suggest. $\endgroup$
    – Mike Shulman
    Commented Feb 26, 2021 at 17:25
  • $\begingroup$ I believe essentially the same approach as in the monoidal case should, using the strictification theorem for bicategories. $\endgroup$
    – varkor
    Commented Sep 25 at 15:32

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