Mark Hovey writes:

What are the standard references for weighted limits and colimits in enriched categories? I know about Borceux, volume 2, chapter 6, but that does not go far enough.

Have a look at Chapter 3 of Max Kelly's book ``The basic concepts of enriched category theory'', LMS lecture note series 64.

More precisely, I want to know how functorial the weighted colimit is in the weight. Given a V-natural transformation F --> F', presumably I get some kind of map from colim_F G to colim_F' G (or the other way around). I would like a reference for this fact and related functoriality facts.

Yes, it is functorial in F, in the way that you have written above.

Presumably the weighted colimit is a bifunctor in the weight and the functor one is taking the colimit of, and presumably this bifunctor has various good properties. Has anybody ever written these down?

Once again, yes. There's quite a lot in the reference above. Kelly writes F*G for what you have called colim_F G, and {F,G} for what you would presumably call lim_F G. He develops results based on the intuition that * is a kind of tensor product, and {-,-} a kind of internal hom, and proves results like the associativity of * and {F*G,H}={F,{G,H}}. Of course this sounds a bit odd, because the F and the G live in different categories, but you can actually make sense of these things. There is even an isomorphism F*G=G*F in the case of colimits in V itself. There's another approach, which probably goes back to Street and Walters, Yoneda structures on 2-categories, J. Algebra 50:350-379, 1978 and has been developed by Street and many others since. In this approach you define a bicategory, often called V-Prof or V-Mod, in which an object is a V-category, and a morphism from A to B, often written A-|->B is a V-functor from A to [B^op,V], and a 2-cell is a natural transformation. (This direction of the 1-cells and 2-cells in this definition is not universally adopted.) Then composition is given by colimit, in the sense that if f:A-|->B and g:B-|->C, then gf:A-|->C is defined by gf(a,c)=g(-,c)*f(a,-). Then associativity (up to isomorphism) of composition in this bicategory is the associaitivity of * referred to above. In fact this bicategory is closed, in the sense that composition has an adjoint, constructed using {-,-}. Steve Lack.