TITLE: What are the higher morphisms between enriched higher categories?
QUESTION [13 upvotes]: This question is about $n$-categories, or perhaps $(\infty,n)$-categories, or ...  My guess is that the answer will not depend sensitively on the model of higher categories, so rather than have me force you to work in my favorite model, I ask only that you say with some precision what model you are using, and that your model is not too strict.
You may assume that I have, and am comfortable with, some chosen symmetric monoidal $n$-category $(S,\otimes)$.  I am trying to understand the notion of "($n$-)category enriched in $S$".  By this I would like to mean the following.  Roughly,  $A$ is enriched in $S$ if $A$ has a set of $0$-morphisms, and between any pair of $0$-morphisms $X,Y \in A$, there is an $S$-object $A(X,Y)$ of 1-morphisms between them.  The composition should be a morphism $A(X,Y) \otimes A(Y,Z) \to A(X,Z)$ in $S$.
I am fairly satisfied that, at least in my example, I can write all of this down explicitly.  Recall that $S$ comes with a chosen object $1 \in S$.  Then the corepresentable functor $S(1,-) : S \to (n-1)\text{-Cat}$ is symmetric monoidal in an essentially unique way.  The usual thing is to use this functor, and define the de-enrichment $A_\delta$ of $A$ to be the $n$-category with hom-$(n-1)$-categories given by $A_\delta(-,-) = S(1,A(-,-))$.  Certainly in my example I can work out this $(n-1)$-category.  (De-enrichment is often a highly lossy operation, and I am OK with that.)
Suppose that $A$ and $B$ are both $S$-enriched categories.  This is where I start to run into trouble.  I understand what an $S$-enriched functor $A \to B$ is.  But I'm having trouble figuring out what is the correct definition of "natural transformation", and in general of the higher morphisms.

Question: Given a symmetric monoidal $n$-category $S$, what is the $(n+1)$-category of $S$-enriched ($n$-)categories?  In particular, what are the higher morphisms?

I recognize that this "$(n+1)$-category of $S$-enriched categories" is likely itself enriched in $S$-enriched categories.  But I am interested simply in writing down its de-enrichment — I'm looking for it just as an $(n+1)$-category.
A final remark: The notion of enrichment in this question is not the same as in n-categories enriched in an (n+1)-category.  That question concerned $n$-categories in which the collection of $n$-morphisms was an object of the enriching category.  I would like the collection of $1$-morphisms to be an object of the enriching category.

REPLY [3 votes]: Another way to get the definition of an $S$-transformation, if you know what an $S$-functor is and also the tensor product of $S$-categories (so $S$ must be symmetric) and $S$ has an initial object preserved by $\otimes$ in each variable, is as an $S$-functor
$$ C \otimes \mathbf{2} \to D$$
where $\mathbf{2}$ is the $S$-category with two objects $a$ and $b$, hom-objects $1$ from $a$ to $a$, $b$ to $b$, and $a$ to $b$, and the initial object of $S$ from $b$ to $a$.