Abstract

If M and V are symmetric monoidal closed categories, then M may carry an enrichment in V, but we show that such an enrichment is essentially the same as a normal morphism M—> V. We make this precise as an equivalence of 2-categories. Next, we show that this equivalence lifts to an equivalence between 2-categories whose objects are, respectively, (1) tensored symmetric monoidal closed V-categories and (2) symmetric monoidal closed adjunctions with right adjoint valued in V. Further, whereas every normal closed functor carries an enrichment, we show also that (1) and (2) are equivalent to a third 2-category whose objects are adjunctions F -| G : M—> V in the 2-category of symmetric monoidal closed V-categories. Along the way, we study change of base for symmetric monoidal V-categories, and we show that the assignment to each symmetric monoidal closed functor its associated enriched functor is part of a 2-functor valued in an op-fibred 2-category of enriched symmetric monoidal closed categories.