Structure of the category of -enriched categories for various contexts
If is a category equipped with a monoidal structure, then Cat has a unit object , and an association sending every -morphism to the a -morphism in the lax slice 2-category :
is really just the underlying category with hom-collections given by .
is the fully faithful two-variable hom-functor from , with defined as the composite in
is the functor from to underlying the enriched functor . This is defined by letting be the composite where is the family of morphisms in defining the enriched functor .
The natural transformation has for its components exactly the maps above: i.e. .
In particular, we recover as when we consider an enriched functor of . The unit identities then hold automatically.
This is in fact a -functor from Cat to . It is not a -functor since if is an enriched natural transformations from an enriched functor to an enriched functor , then , the underlying natural transformation from to satisfies instead of that the definition of requires. Nevertheless, if we do define the codomain of the association to have only the objects and -morphisms as in the above triangles, and define -morphisms to satisfy the correct naturality condition (rather than the lax slice 2-category condition), then we still have a -category and the association is a -functor that is fully faithful on -morphisms, and faithful on -morphisms.
Two remarks are in order. First, something is deeply wrong here since this -category we have had to define arises out of the desire to make things easy and understandable, as opposed to natural. Second, the failure of the association to be faithful on -morphisms means that an enrichment of a functor consists of more than just giving a natural transformation from to . But more significantly, this seems to suggest that the monoidal structure on is not captured by any structure on the -category Cat. This state of affairs is remedied in any closed category context.