Dear category theorists, I have a question concerning the paper "On Closed Categories of Functors" from Brian Day (By the way, this is an excellent paper). Let V be a symmetric monoidal closed category and C a small V-category. The (ordinary) category [C,V] of V-functors admits the sturcure of a V-category in a canonical way. A symmetric monoidal V-category is the enriched analogue of a symmetric-monoidal structure on an ordinary category, i.e. all the structure morphisms are V-morphisms and the coherence conditions are fullfilled. The underlaying category of a symmetric monoidal V-category admits the structure of an ordinary symmetric monoidal category. Brian Day constructs a symmetric monoidal closed structure ([C,V],@,E) on the V-category of V-functors [C,V] for some cases [3.3, 3.6], e.g. if (C,*,e) is a symmetric monoidal V-category [4.1]. The underlaying *category* [C,V] of V-functors admits a closed symmetric monoidal structure from the enriched one by taking the underlaying functor of each V-functor, the underlaying natural transformation of each V-natural transformation. Because a closed symmetric monoidal category is canonically enriched over itself, the category [C,V] gets a [C,V] enrichment in this way. My question is: What does this [C,V]-enrichment of [C,V] have to do with the V-enrichment of [C,V]? Suppose C have a terminal object t. One gets a evaluation functor Ev_t:[C,V]-CAT-->V-CAT. Is this the connection between the two enrichments? Thank you in advance for any help. Tony