Never mind, now I see you just wanted a clean up my formulation of \\(\mathbb{ME}\\) now that I reread this.

I think we both see now how to embed \\(\mathbf{Grp}\\) into \\(\mathbb{ECM}\\).

It looks like \\(\mathbb{ECM}\\) is a subcategory of \\(\mathbb{ECP}\\):

> - An object is a pair \\((\mathcal{V}, \mathcal{X})\\) where \\(\mathcal{V}\\) is a monoidal preorder and \\(\mathcal{X}\\) is an enriched category over \\(\mathcal{V}\\).

>

> - A morphism \\((\mathcal{V}, \mathcal{X}) \to (\mathcal{U}, \mathcal{Y})\\) consists of a pair \\(\left(\hat\phi\colon \mathcal{V}\to \mathcal{U}, \,\,\phi \colon \text{Ob}(\mathcal{X})\to \text{Ob}(\mathcal{Y})\right)\\) where \\(\hat\phi\\) is a monoidal preorder map and we have \\(\hat\phi(\mathcal{X}(a,b)) \leq \mathcal{Y}(\phi(a),\phi(b))\\).

Now if we restrict ourselves now to \\(\hat{\phi} = id\\) and \\(\mathcal{V} = \mathbf{Cost}\\) then we recover just the \\(\mathbf{Cost}\\)-categories and functors.

I see why you say \\(\mathbb{ECM}\\) feels more like an enriched category theory construction.

I think we both see now how to embed \\(\mathbf{Grp}\\) into \\(\mathbb{ECM}\\).

It looks like \\(\mathbb{ECM}\\) is a subcategory of \\(\mathbb{ECP}\\):

> - An object is a pair \\((\mathcal{V}, \mathcal{X})\\) where \\(\mathcal{V}\\) is a monoidal preorder and \\(\mathcal{X}\\) is an enriched category over \\(\mathcal{V}\\).

>

> - A morphism \\((\mathcal{V}, \mathcal{X}) \to (\mathcal{U}, \mathcal{Y})\\) consists of a pair \\(\left(\hat\phi\colon \mathcal{V}\to \mathcal{U}, \,\,\phi \colon \text{Ob}(\mathcal{X})\to \text{Ob}(\mathcal{Y})\right)\\) where \\(\hat\phi\\) is a monoidal preorder map and we have \\(\hat\phi(\mathcal{X}(a,b)) \leq \mathcal{Y}(\phi(a),\phi(b))\\).

Now if we restrict ourselves now to \\(\hat{\phi} = id\\) and \\(\mathcal{V} = \mathbf{Cost}\\) then we recover just the \\(\mathbf{Cost}\\)-categories and functors.

I see why you say \\(\mathbb{ECM}\\) feels more like an enriched category theory construction.