The word I was looking for was 'function'. Sorry.

You have defined a function on a certain subset of \$$\mathcal{V}\$$, specifically the subset \$$\langle\mathcal{V}_X \rangle\subset \mathcal{V}\$$ defined to be the submonoid generated by the elements of \$$\mathcal{V}_X :=\\{\mathcal{X}(a,b) \,|\, a,b\in Ob(\mathcal{X})\\}\$$.

Now, \$$\langle \mathcal{V}_X \rangle\$$ is a monoid (by construction) and is the smallest monoid that \$$\mathcal{X}\$$ can be considered to be an enriched category over. In fact, as you've probably pointed out somewhere, \$$\langle \mathcal{V}_X \rangle\$$ is a group.

So what you've constructed (I think!) as a morphism \$$\mathcal{X}\to\mathcal{Y}\$$ in ME is a function \$$\phi\colon\text{Ob}(\\mathcal{X}) \to \text{Ob}(\\mathcal{Y})\$$ and a monoid/group homomorphism \$$\hat\phi\colon \langle \mathcal{V}_X \rangle\to \langle \mathcal{U}_Y \rangle\$$, such that \$$\hat\phi(\mathcal{X}(a,b)) = \mathcal{Y}(\phi(a),\phi(b))\$$ .