Spahn Ad(K)

Let $K$ be a $2$-category, then adjunction $\dashv$ is a relation on $K_1$ (the $1$-cells of $K$). The relation is linear and composition makes $K_1$ into a pomagma $Ad(K)$ whose object set is $K_1$ and $hom(l,r)=(l\dashv r)$ if $l\dashv r$ and empty otherwise.

We can also consider $Ad(K)$ as a “$2$-twisted $2$-category” where we have $2$-morphisms between antiparallel $1$-cells and whose level $2$ is thin.

$\dashv$ satisfies

$$if\;(l\dashv r\dashv s)\; then\; (rl\dashv rs)\;and\;(lr\dashv sr)$$