categories: question about enriched category theory

[Gaunce Lewis <lglewis@syr.edu>]

Assume that V is a symmetric monoidal closed category and that A and B are 
categories enriched over V which have tensors.  Let me denote the tensor of 
an object v of V and an object a of A by

v \ten a

Assume also that F is a functor from the underlying ordinary category A_0 
of A to the underlying category B_0 of B.  If F were enriched over V, then 
there would be a natural map

f : v \ten Fa --> F(v \ten a)

describing the behavior of F on tensors.

However, assume only that F is a functor on the underlying categories.  It 
seems to me that, if there is a well-behaved natural map f of the above 
form for all v in V and a in A, then F ought to be enriched over V.  It is 
easy to construct from f the map that ought to be the enrichment for 
F.  The trick is to decide what properties f must have in order to ensure 
that the putative enrichment really works.  Is this written up somewhere?

Along the same lines, suppose now that F and G are enriched functors from A 
to B with the associated maps

f : v \ten Fa --> F(v \ten a)

and

g : v \ten Ga --> G(v \ten a)

describing their behavior on tensors.  Assume also that t is an ordinary 
natural tranformation between the ordinary functors F_0 and G_0 underlying 
F and G.  There is an obvious diagram relating t, f, and g, and it seems 
that this diagram ought to commute if t is an enriched natural 
transformation.  In fact, it seems that the commutativity of this diagram 
ought to be equivalent to t being enriched over V.  Is this written down 
anywhere?

Thanks for any help on this,
Gaunce