Well, my point wasn't that we have a functor, but rather that was this is a more category theoretic way to express what the category \\(\mathbb{ME}\\) is.

Then, it makes it straight forward to see that we have the functor that you were claiming.

Having said that, \\(\mathbb{ME}\\) doesn't quite feel like a natural category to me, unless you can give me a place where it crops up naturally.

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))\\).

Then you still get the functor \\(\mathcal{G}\colon \textbf{Grp}\to \mathbb{ECM}\\), but I think that I can now construct two natural left inverses.

My point here is that \\(\mathbb{ECM}\\) feels more like an enriched category theory construction, and easily generalises to enriching over arbitary monoidal categories. However, that's not to say you won't have a naturally occurring mathematical situation in which you want to deal with \\(\mathbb{ME}\\).