For every $P,Q, x : P \to X, y : Q \to Y$ we consider the diagram of comma objects, where $x \tdown_\vf y$ is a strict pullback, and $x\btdown y : x \tdown_\vf y \to Ux\downarrow Vy$ [the dotted arrow, TN] is the functor induced by the universal property of said comma object.

If $\pi_0(x\btdown y)$ is a bijection we say that $\vf$ is exact at $\smat{x \\ y}$ and we write $\smat{x \\ y}\vDash \vf$. So, $\vf$ exact equals the following condition ($\pi_0\tdown$): for all $P,Q, x : P \to X, y : Q \to Y$, we have $\smat{x \\ y}\vDash\vf$.