Thank you for [the answer](https://forum.azimuthproject.org/discussion/comment/19042/#Comment_19042), John!
It does make a lot of sense!
I will try to remind you to discuss matrix multiplication for categories enriched over monoidal categories.
Are there any available online resources on this topic?

I wonder, however, what is the precise connection between quantale matrix multiplication and quantale-enriched categories?
Is it true that given a quantale \\(\mathcal{V}\\) and two \\(\mathcal{V}\\)-enriched categories \\(\mathcal{X}\\) and \\(\mathcal{Y}\\) then their matrix multiplication \\(\mathcal{X}\*\mathcal{Y}\\) is also a \\(\mathcal{V}\\)-enriched category?
(I see that in section 2.5.3 of the book the repeated matrix multiplication is used to obtain the desired quantale-enriched categories.)