### Abstract

B ́enabou pointed out in 1963 that a pair f ⊣ u : A → B of adjoint functors induces a monoidal functor [f, u] : [A, A] → [B, B] between the (strict) monoidal categories of endofunctors. We show that this result about adjunctions in the monoidal 2-category Cat extends to adjunctions in any right-closed monoidal 2-category V, or more gen- erally in any 2-category A with an action ∗ of a monoidal 2-category V admitting an adjunction A(T ∗ A, B) ∼= V(T, ⟨A, B⟩); certainly such an adjunction exists when ∗ is the canonical action of [A, A] on A, pro- vided that A is complete and locally small. This result allows a concise and general treatment of the transport of algebraic structure along an equivalence.

Original language | English |
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Title of host publication | Galois theory, Hopf algebras, and semiabelian categories |

Editors | George Janelidze, Bodo Pareigis, Walter Tholen |

Place of Publication | Providence, R.I. |

Publisher | American Mathematical Society |

Pages | 319-340 |

ISBN (Print) | 9780821832905 , 9780821871478 |

Publication status | Published - 2004 |

Externally published | Yes |

### Publication series

Name | Fields Institute Communications |
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Publisher | American Mathematical Society |

Volume | 43 |

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## Cite this

Lack, S. (2004). Monoidal functors generated by adjunctions, with applications to transport of structure. In G. Janelidze, B. Pareigis, & W. Tholen (Eds.),

*Galois theory, Hopf algebras, and semiabelian categories*(pp. 319-340). (Fields Institute Communications; Vol. 43). Providence, R.I.: American Mathematical Society.