# Monoidal functors generated by adjunctions, with applications
to transport of structure

### G.M. Kelly and Stephen Lack

Bénabou pointed out in 1963 that a functor *f:B-->A* with
a right adjoint *u* 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 generally 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**, 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.

Click here to download a gzipped
postscript file of the complete paper.

Steve Lack
Last modified: Mon Dec 23 17:53:13 EST 2002