In a 2-category $\mathbb K$ with a Yoneda structure, an admissible square $\vf : \esq{S}{U}{T}{V}$ is exact according to BC# if and only if the diagram
exhibits $B(U,1)\circ V$ as the left extension of $Y_X\circ S$ along $T$. This is condition PP#
If the Yoneda structure is uniform conditions BC# and PP# are equivalent for the squares $\vf$ such that $U$ is admissible.
Relative adjunctions and, more in general, absolute extensions are (as in the case of $\Cat$) exact squares in the sense of BC#.