The fact that $\vf$ is exact is equivalent, through ZZ, to a condition LI of local initiality or to a condition LF of local finality:

• (local initiality) $x\in X_0$, $x\downarrow S \to Ux\downarrow V$ is initial [this is the functor sending $(x\to Sa)$ into $(Ux \to USa \xto{\vf} VTa)$, TN];
• (local finality) $y\in Y_0$, $T\downarrow y \to U\downarrow Vy$ is final [this is the functor sending $(Ta\to y)$ into $(USa \xto{\vf} VTa \to Vy)$, TN].