A square $\vf : US \To VT$ in $\Cat$ is exact if and only if, for every pointwise left extension, (cf. (MacLane, 1971) (Street, 1974)), the composite triangle remains a pointwise left extension.
In fact, consider $G : Y \to Z$. Using 1·f [001V]) for $\vf$, we have $\cate{Nat}(FS,GT)\cong \cate{Nat}(Fd_0, Gd_1,G)$, and by the assumption made on $\beta$, we have that the latter is isomorphic to $\cate{Nat}(RV,G)$. Hence, $(R*\vf) \c (\beta * S) : FS \To RVT$ is an extensions, and moreover it is pointwise, thanks to Theorem 1.6.
For the converse, 1·c·b [001P]) means that for every $x\in X_0$ the diagram (with $i_x(y) : X[x,y] \to B[Ux, Uy]$ the action of $U$ on morphisms) is a pointwise exension. To conclude, it remains to check that $B[Ux, \firstblank]$ is the pointwise extension of $X[x,\firstblank]$ along $U$.