> Now you've got a subset of \\(\mathcal{V}\\) that we can call \\(\mathcal{V}_X\\) which is \\(\\{\mathcal{X}(a,b) \,|\, a,b\in Ob(\mathcal{X})\\}\\). Given any element of \\(\mathcal{V}\\) which is a product of elements of \\(\mathcal{V}_X\\) you associate an element of \\(\mathcal{U}\\) which is independent of how it is expressed as a product. There's a more 'high-level' way of saying this, which will make what you've written easier to relate to other ideas about functors between enriched categories. I suspect you know this higher level description and use it when you bring groups in to the picture.

Hmm... I am not sure what word you are looking for.

It's charitable to assume I know things. I sort of have the "jack of all trades, master of none" going for me, only it's more like "jack of measure-zero trades, master of \\(-\infty\\)" where \\(-\infty < 0\\) is the thing Chris Upshaw invented in [the other thread](https://forum.azimuthproject.org/discussion/comment/18568/#Comment_18568).