I should warn people that I've fixed my convention concerning the cap: now it's

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

So, the above discussions of the cap now look 'backwards', because they were based on the convention where \$$\cap_X \colon \textbf{1} \nrightarrow X^{\text{op}} \times X \$$. This is nobody's fault but mine! Luckily, it's easy to adjust these arbitrary conventions.