$\require{AMScd}$Are there references for a construction of the enriched slice category of $\mathcal A \in \mathcal{V}\text{-Cat}$? A reasonable definition should be 


1. Fix an object $a\in\mathcal A$ and let the objects of ${\cal A}/a$ be the set ${\cal V}(J, {\cal A}(x,a))$ for all $x\in\cal A$. $J$ is the monoidal unit of $\cal V$.
2. Let ${\cal A}/a(p,q)$ be the $\cal V$-object resulting from the pullback
$$
\begin{CD}
 P @>>> {\cal A}(x,y) \\
 @VVV {}@VVq\_\*V\\
J @>>p> {\cal A}(x, a)
\end{CD}
$$
for $p : x\to a, q : y \to a$ and $J$ the monoidal unit.


Does it work for every $\cal V$?



Enriched slice categories
There is at least one sense in which this "works". Namely, it is the comma object in the 2-category $\mathcal{V}$-Cat of $\mathrm{Id} : \mathcal{A} \to \mathcal{A}$ over $[a] : \mathcal{J} \to \mathcal{A}$, where $\mathcal{J}$ is the unit $\mathcal{V}$-category with one object and $J$ as its hom-object. It's true that it doesn't capture as much information in the enriched case as it does in the ordinary case, but given that it has this universal property I don't know what else one might mean by "enriched slice category".


