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.

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.