Anindya wrote:

> However in this case we could slightly vary the definition of a \\(\mathcal{V}\\)-enriched category to use right-to-left composition instead of left-to-right composition.

Hmm, as mentioned in my previous comment I don't see a left/right asymmetry built into the definition of enriched category:

> **Definition.** A **\\(\mathcal{V}\\)-enriched category** \\(\mathcal{X}\\) consists of two parts, satisfying two properties. First:

> 1. one specifies a set \\(\mathrm{Ob}(\mathcal{X})\\), elements of which are called **objects**;

> 2. for every two objects \\(x,y\\), one specifies an element \\(\mathcal{X}(x,y)\\) of \\(\mathcal{V}\\).

> Then:

> a) for every object \\(x\in\text{Ob}(\mathcal{X})\\) we require that

> \[ I\leq\mathcal{X}(x,x) .\]

> b) for every three objects \\(x,y,z\in\mathrm{Ob}(\mathcal{X})\\), we require that

> \[ \mathcal{X}(x,y)\otimes\mathcal{X}(y,z)\leq\mathcal{X}(x,z). \]

The interesting thing is that you can interpret \\(\mathcal{X}(x,y)\\) as 'morphisms from \\(x\\) to \\(y\\)' or 'morphisms from \\(y\\) to \\(x\\)' and the definition makes sense either way!

Are you suggesting that we could switch to using

\[ \mathcal{X}(y,z)\otimes \mathcal{X}(x,y) \leq\mathcal{X}(x,z)? \]

We could, but you'll notice this isn't just a left/right reflection of the usual definition: I could explain the difference to someone who can't tell the difference between left and right! In the usual definition, the two \\(y\\)'s here are next to each other:

\[ \mathcal{X}(x,y)\otimes\mathcal{X}(y,z) \]

while in the modified definition they aren't.