[001P] 1·c·b (BC'' criterion).

Since coends can be computed as colimits (MacLane, 1971), $\vf$ exact is equivalent to the fact that for all $x\in X_0, y\in Y_0$, the maps $\tilde\vf_{x,y}(a)$ induce an isomorphism

$$ \hat\vf_{x,y} : \colim\left( T\downarrow y \to A \xto{X[x,S\firstblank]} \Set \right) \overset\cong\to B[Ux, Vy]. $$

Symmetrically, we obtain the equivalent condition: for all $x\in X_0, y\in Y_0$, the maps $\tilde\vf_{x,y}(a)$ induce an isomorphism

$$ \check\vf_{x,y} : \colim\left( (x\downarrow S)^\op \to A^\op \xto{X[T^\op\firstblank,y]} \Set \right) \overset\cong\to B[Ux, Vy]. $$