Hi, David! Most of your calculations look good to me; I added a bit of extra space to make them easier for me to read.

But you seem to be using some formulas for \$$\cap_X\$$ and \$$\cup_X\$$ that I don't know. These guys are not enriched functors; they're enriched profunctors, so you can't simply apply them to an object and get another object. For example \$$\cap_X\$$ is an enriched profunctor

$\cap_X \colon \textbf{1} \nrightarrow \mathcal{X} \otimes \mathcal{X}^{\text{op}}$

but you seem to be applying it to the object \$$1 \in \mathbf{1}\$$ and getting

$\bigvee_{y \in \mathcal{X}} y^{\text{op}} \otimes y$

I don't know what this expression even means! \$$y\$$ is an object in \$$\mathcal{X}\$$, \$$y^{\text{op}}\$$ is a perfectly fine name for the corresponding object in \$$\mathcal{X}^{\text{op}}\$$ (which, remember, has the same set of objects. I don't know what it means to tensor an object of \$$\mathcal{X}\$$ and one of \$$\mathcal{X}^{\text{op}}\$$, but I can make something up: the objects of \$$\mathcal{X} \otimes \mathcal{X}^{\text{op}}\$$ are pairs \$$(x,y)\$$ consisting of an object in \$$\mathcal{X}\$$ and one in \$$\mathcal{X}^{\text{op}}\$$; if we were in a certain mood we could write such a pair as \$$x \otimes y\$$. The worst problem comes at the end: I don't what it means to take the join of a bunch of such tensor products. I can take joins in a poset, like \$$\mathcal{V}\$$, but \$$\mathcal{X} \otimes \mathcal{X}^{\text{op}}\$$ is not a poset.

It's possible you're doing something correct, but I'd need some more explanation to understand what

$\bigvee_{y \in \mathcal{X}} y^{\text{op}} \otimes y$

means and why we should think of \$$\cap_X\$$ as sending \$$1 \in \mathbf{1}\$$ to this thing.