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))\\) .

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))\\) .