We will suppose, for the sake of simplicity, that everything is happening within a fixed 'metacategory' $\textbf{SET}$ of sets and functions. So, from now on, a 'category' just means a category object in $\textbf{SET}$ - i.e. a small category.

Let $\mathscr{V}$ be a monoidal category. A $\mathscr{V}$-enriched category $\mathscr{C}$ consists of:

- Objects: A set Ob($\mathscr{C}$).
- Morphisms: For each pair of $\mathscr{C}$-objects $(X, Y)$, a $\mathscr{V}$-object Hom$(X, Y)$.
- Composition: For each triple of $\mathscr{C}$-objects $(X, Y, Z)$, a $\mathscr{V}$-morphism $\circ$ : Hom$(X, Y)$ $\otimes$ Hom$(Y, Z)$ $\rightarrow$ Hom$(X, Z)$.
- Identities: For each $\mathscr{C}$-object $X$, a $\mathscr{V}$-morphism id$_X$: $I$ $\rightarrow$ Hom$(X, X)$ (where $I \in \mathscr{V}$ is the unit of $\otimes$).

This data is then subject to the usual associativity and unitality axioms which are expressed via the commutativity of certain diagrams in $\mathscr{V}$. From this enriched category, we can extract an underlying category $\mathscr{C}_0$ by defining $\mathscr{C}(X, Y) = \mathscr{V}(I, \text{Hom}(X, Y))$.

My question is about if this is reversible - namely, can we define a $\mathscr{V}$-enriched category to be a category $\mathscr{C}$ equipped with a 'hom-functor' to $\mathscr{V}$? I'm having some trouble finding a reference for this but it seems like there should be a fairly obvious definiton. A $\mathscr{V}$-atlas on a category $\mathscr{C}$ consists of:

- Morphisms: A functor Hom: $\mathscr{C}^{op} \times \mathscr{C} \rightarrow \mathscr{V}$.
- Composition: For each triple of $\mathscr{C}$-objects $(X, Y, Z)$, a $\mathscr{V}$-morphism $\circ$ : Hom$(X, Y)$ $\otimes$ Hom$(Y, Z)$ $\rightarrow$ Hom$(X, Z)$.
- Parametrisation: For each pair of $\mathscr{C}$-objects $(X, Y)$, an isomorphism $\eta: \mathscr{C}(X, Y) \xrightarrow{\sim} \mathscr{V}(I, \text{Hom}(X, Y))$ such that for all $X \xrightarrow{f} Y \xrightarrow{g} Z$ in $\mathscr{C}$, $\eta(g \circ f) = \eta(g)\circ\eta(f)$ (where on the left we have compositon in $\mathscr{C}$ and on the right we have composition in $\mathscr{V}$).

I'm unsure though if this gives associativity and unitality as in the usual defintion of a $\mathscr{V}$-enriched category, or if we only get associativity and unitality for $I$-shaped elements of the hom-objects. Could this be remedied by just requiring the associativity and unitality laws to hold as in the usual definiton? Any help or references would be much appreciated.