TITLE: Enriched Cauchy completions and underlying categories
QUESTION [5 upvotes]: The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] \simeq [\overline{C}^{op}, Set]$, etc...
There's also a notion of Cauchy completion for enriched categories, my questions are about it:
1 - Let $X$ be a $V$-enriched category (where $V$ is a closed symmetric monoidal category with all limits and colimits), what properties does its enriched Cauchy completion $\overline{X}$ satisfy? Like is there an equivalence $[X^{op}, V] \simeq [\overline{X}^{op}, V]$, etc?
2 - What can be said about the underlying categories of $X$ and $\overline{X}$ ($X_0$ and $\overline{X}_0$)?? Is $\overline{X}_0$ the ordinary Cauchy completion of $X$? Do we have $[X_0^{op}, Set] \simeq [\overline{X}_0^{op}, Set]$, etc?
I just wanted to ask before I go about trying to answer this myself.

REPLY [3 votes]: It works the other way around --- you should have tried to answer your question by yourself before you posted it here :-)

Yes, there is an equivalence $[X^{op}, V] \simeq [\overline{X}^{op}, V]$. You may find more details in "Basic Concepts of Enriched Category Theory" by M. Kelly (Chapter 5.5).
No. The name "Cauchy completion" has been chosen to suggest that it is a  generalization of the usual concept of Cauchy completion for metric spaces. Indeed, if you take a generalized Lawvere metric space $X$, then the usual completion of $X$ under Cauchy sequences coincide with the Cauchy completion of $X$, when $X$ is thought of as a category enriched over poset $[0, \infty]$ with monoidal structure $\langle 0, {+}\rangle$. On the other hand, the underlying category of $X$ is always Cauchy complete in the sense of ordinary categories.