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.