$\mathcal{V}$-naturality in enriched category theory Let $\mathcal{V}$ be a monoidal category, in section 1.2 of "Basic concepts of enriched category theory" (http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) Max Kelly introduces the terms "$\mathcal{V}$-category", "$\mathcal{V}$-functor" and "$\mathcal{V}$-natural transformation" that make $\mathcal{V}$-CAT a 2-category. In section 1.7 he says:

... all the families of maps canonically associated to $\mathcal{V}$, or
  to a $\mathcal{V}$-Category $\mathcal{A}$, or to a
  $\mathcal{V}$-functor T, or to a $\mathcal{V}$-Natural $\alpha$, such as $a: (X
\otimes Y) \otimes Z \rightarrow X \otimes ( Y \otimes Z)$, or 
  $e: [Y,Z] \otimes Y \rightarrow Z$, or $M: \mathcal{A}(B, C) \otimes
\mathcal{A}(A, B) \rightarrow \mathcal{A}(A, C)$ 
  or $T:\mathcal{A}(A,B) \rightarrow \mathcal{B}(TA, TB)$, 
  or $\alpha: I \rightarrow \mathcal{B}(TA, SA)$, are themselves $\mathcal{V}$-natural in every variable...

What does he mean by these families of maps "are themselves $\mathcal{V}$-Natural in every variable"? I thought the term $\mathcal{V}$-Naturality referred to $\mathcal{V}$-natural transformations $\alpha: T \rightarrow S: \mathcal{A} \rightarrow \mathcal{B}$ between $\mathcal{V}$-functors $T, S: \mathcal{A} \rightarrow \mathcal{B}$ (for $\mathcal{V}$-categories $\mathcal{A}$ and $\mathcal{B}$).
 A: I believe you fall in the same problem I had while reading Kelly's book.
As user54748 suggested in a comment all the natural functors (the tensor product, internal hom, etc) lift to enriched $\mathcal V$-functors between the corresponding $\mathcal V$-categories. Similarly all the natural (apologize for the game of words) natural transformations between these functors lift to $\mathcal V$-natural transformations.
(A note: by the functors/natural transformations lift I mean that they are the images of $\mathcal V$-enriched functors/natural transformations through the underlying category functor, that should be named  $\mathcal V_o \colon \mathcal V\text{-}\mathbf{Cat} \to \mathbf{Cat}$ if I remember correctly).
For what I remember, after the chapter on symmetric monoidal closed categories, Kelly basically exploits the categorical isomorphism between the category $\mathcal V$ and the underlying category to the $\mathcal V$-enriched category $\mathcal V$ (the category where $\text{hom}(A,B)=\mathcal V[I,[A,B]]$), and so it identifies $\mathcal V$-enriched functors and natural transformations with their not enriched counterparts.
That caused to me quite some problems, because in my personal opinion such identification is an abuse of notation, and the worst part is that the reader is not even warned (or at least I don't remember any warning).
Hope that this (not so short answer) could be of help (or at least that knowing that other people had the same problem could be of some confort).
