We denote the comma square of the pair $U,V$, and $\bar\vf : A \to U\downarrow V$ the only morphism such that $a * \bar\vf = \vf$.

For every $Z, P : X \to Z, Q : Y \to Z$ we have where $\sum_{P,Q}\tilde\vf$ and $\sum_{P,Q}\bar\vf$ denote the composition functors with $\tilde\vf$ and $\bar\vf$. [The correspondences are defined as follows: TN]