Proposition. Uniqueness of restrictions in a double category [mde-K81D]

Let the following be two cartesian cells in a double category,

then \(p \cong p'\).

Proof.  [#249]

The cartesian property of \(\phi \) gives us \(\phi ' = \frac {i}{\phi }\) for a unique globular cell \(i : p' \to p\), and the cartesian property of \(\phi '\) gives us that \(\phi = \frac {i^{-1}}{\phi '}\) for a unique globular cell \(j: p \to p'\). With similar reasoning starting from \(\phi '\), we see that \(i\) and \(j\) are inverses.