I know this question may seem nonsensical at first but let me exlain what i have in mind:


In enriched category theory we define categories enriched over a monoidal category $(\mathcal{V},\otimes, I)$. An enriched category then is given by a set/class of objects $\mathcal C$ and a rule assigning to every pair $X,Y$ of such objects a hom-object $[X,Y]$. Furthermore we define composition and identities using $\otimes$ and $I$, remodelling the definitions of usual category theory.


**Now for the question**: Can we go the other way around?


Let's stick to internal homs for the beginning: Given a category $\mathcal V$ ; can say what additional data turn a functor $$[-,-]:\mathcal{V}^{\mathrm{op}}\times\mathcal V\to \mathcal V$$ into something like an internal hom?


In the case of $[X,-]$ having a left adjoint $-\otimes X$ for every $X$, these additional data should result in $(\mathcal V,\otimes)$ becomming a closed monoidal category with internal hom isomorphic to $[-,-]$.



Definition of enriched caterories or internal homs without using monoidal categories.
This is exactly the notion of a [closed category](http://en.wikipedia.org/wiki/Closed_category). See Eilenberg and Kelly's article in the 1965 La Jolla proceedings (Springer 1966). I think they also describe categories enriched in a closed category.


