I've thrown together a (rather poor, compared to yours) diagram illustrating the distinction I'm trying to make. On the right side is the situation as we have it now, with a reaction \$$w \le x \otimes y \otimes z\$$ assuming associativity.

![](https://i.imgur.com/W7fTaPH.png)

If we don't have associativity, then there are two possible reactions we might have meant. Let's suppose we want \$$w \le (x \otimes y) \otimes z\$$ to be our reaction. We need to be able to distinguish between \$$(x \otimes y) \otimes z\$$ and \$$x \otimes (y \otimes z)\$$, since the latter isn't a valid input to our reaction. So we have to represent the input differently; in this case, with explicit \$$\otimes\$$ combiner nodes and reducing all reactions to single-input single-output representations.

In other words, if you replaced \$$\le\$$ with \$$\otimes\$$ in this diagram (and added the missing output edges), it would be completely accurate:

![](http://aether.co.kr/images/monoidal_preorder_associativity.svg)

But since we bake associativity into our rules for working with string diagrams (namely, allowing multi-input multi-output relations), we never actually need the \$$\otimes\$$ node. It's an entirely redundant symbol, since it's been incorporated into the representation of our reactions.