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?
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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](https://mathoverflow.net/questions/102492/n-categories-enriched-in-n1-categories). 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.



What are the higher morphisms between enriched higher categories?
For $F,G:C\to D$, an $S$-enriched transformation $F\to G$ will consist of, for each $x\in C$, a morphism $1\to D(F x, G x)$ in $S$, together with for each $x,y\in C$, an equivalence between the composites
$$ C(x,y) \to D(F x, F y) \to D(F y, G y) \otimes D(F x, F y) \to D(F x, G y)$$
and
$$ C(x,y) \to D(G x, G y) \to D(G x, G y) \otimes D(F x, G x) \to D(F x, G y)$$
plus some higher coherence cells until you run out of room. Does that help?


