> What feels more natural is the following which I'll call \\(\mathbb{ECM}\\) (rather than \\(\mathbb{ME}2\\)).

>

> - An object is a pair \\((\mathcal{V}, \mathcal{X})\\) where \\(\mathcal{V}\\) is a monoid 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 monoid map and we have \\(\hat\phi(\mathcal{X}(a,b)) = \mathcal{Y}(\phi(a),\phi(b))\\).

Isn't \\(\mathbb{ECM}\\) isomorphic to the category of groups?

>

> - An object is a pair \\((\mathcal{V}, \mathcal{X})\\) where \\(\mathcal{V}\\) is a monoid 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 monoid map and we have \\(\hat\phi(\mathcal{X}(a,b)) = \mathcal{Y}(\phi(a),\phi(b))\\).

Isn't \\(\mathbb{ECM}\\) isomorphic to the category of groups?