Since every bimodule can be written, in $\Cat$, in the form $P^o\otimes Q$, as a consequence of Yoneda lemma, $\tilde\vf$ is invertible if and only if the $\sum_{P,Q}\tilde\vf$ are bijective so that $\vf$ exact equals the following condition: for all $Z$, $P : X\to Z, Q: Y\to Z$, the function

is a bijection.

There is a similar criterion H$^\op$ with co-comma squares. A 2-category $\mathbb{K}$ that is representable and co-representable, where H $\iff$ H$^\op$ is called symmetric. This is the case of $\Cat$.