I am just starting to learn about enriched categories, so excuse me if I am asking something trivial.
Suppose $\mathcal{C}$ is a $\mathcal{V}$-enriched category $\mathcal{C}$, with $\mathcal{V}$ very rich in structure, for example $\mathcal{V}$ could be the category of $R$-modules for a commutative ring $R$. If the underlying category of $\mathcal{C}$ is cocomplete, how close is $\mathcal{C}$ to being cocomplete as an enriched category? Is there a general theory that answers this question?
Yes. There's no guarantee of cocompleteness just from that for the underlying ordinary category. But if you assume that $\mathcal{C}$ is cotensored over $\mathcal{V}$, then the existence of colimits in the underlying category implies the existence of conical colimits in $\mathcal{C}$, and if $\mathcal{C}$ is tensored, every colimit can be written in terms of conical colimits and tensors, so that $\mathcal C$ is cocomplete. This result appears near the end of Chapter 3 of Kelly's enriched category theory monograph.