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.