I am sorry if this is a trivial matter, but I was unable to find a reference:
Let $\cal{V}$ be a Benabou-cosmos. What is the definition of an initial object in a $\cal V$-category $\cal C$?
From the context of $\mathsf{Ab}$-enriched and $\mathsf{Cat}$-enriched categories it seems like the condition is $\mathcal{C}(I,X) \cong *$ for all $X \in \cal C$, where $*$ denotes the terminal object in $\cal V$. But I am unsure (the underlying category is defined in terms of morphisms $1 \rightarrow C(X,Y)$ and not $* \rightarrow C(X,Y)$...). So I kindly ask for a reference. Thank you very much.
Kelly doesn't explicitly speak about enriched terminal objects in his book, but about weighted limits and the special case of conical limits. If define a terminal object $1$ to be a limit over the empty diagram (in which case there is a unique weight), you get the condition that all hom objects $\mathrm{hom}(A,1)$ are terminal in $\mathcal{V}$, as you suspected.