> 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).