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