Monoidal functors generated by adjunction, with applications to transport of structure

Benabou pointed out in 1963 that a pair f --l u : A -> B of adjoint functors induces a monoidal functor If, u) : [A, A) -> [B, B) between the (strict) monoidal categories of endofunctors. \Ve 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 generally in any 2-category A with an action * of a monoidal 2-category V admitting an adjunction A(T * A, B) ~ VeT, (A, B})j certainly such an adjunction exists when * is the canonical action of [A, A) on A, provided that A is complete and locally small. This result allows a concise and general treatment of the transport of algebraic structure along an equivalence