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.