> Sorry to be so fierce: I'm just playing the teacher, and one of the teacher's jobs is to get people to talk the official standard way, so they can communicate. I figured that if you, one of the best students, hadn't yet absorbed how to talk about enriched categories, I really need to emphasize how it's done.

Thank you for your kind words John.

I have been learning an awful lot from you and the other members on this forum.

I believe I have a handle on enriched categories now!

> If \$$\mathcal{V}\$$ is discrete these inequalities are equations, so we get
>
> $I = \mathcal{X}(a,b) \otimes \mathcal{X}(b,a)$
>
> which means that the subset
>
> $\{ \mathcal{X}(a,b) : a,b \in \mathrm{Ob}(\mathcal{X}) \} \subseteq \mathcal{V}$
>
> is actually a group contained in the monoid \$$\mathcal{V}\$$.

Not exactly.

The set \$$\\{ \mathcal{X}(a,b) : a,b \in \mathrm{Ob}(\mathcal{X}) \\}\$$ may not be closed under \$$\otimes\$$.

It *generates* a group, however.