change of base and co/limit preservation

Let F: V -> W be a monoidal functor, and F*: VCat -> WCat be the change of base; suppose T: A->B is a V-functor which preserves a class of weighted co/limits {(j_i: K ->A, w_i: K -> V)}. What is the correct notion of preservation for F*T; i.e. what is the corresponding class of W-co/limits? It seems reasonable to be {(F*(j_i),F*(w_i)}, but I have not found a good reference for this. Under what conditions does F* "preserve co/limit preservation"? There seem to be some interesting questions here. Has this been studied in any depth? Thank you. Sincerely, Christian Williams [For admin and other information see: http://www.mta.ca/~cat-dist/ ]