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}\$$.