Note that we always have the cartesian cell

Then from uniqueness of restrictions in a double category we have \(M \cong Z(1,F) \odot N \odot W(G,1)\). Conversely, we claim that

is cartesian, which is easily shown with string diagrams. We build the unique cell factoring \(\beta \) by composing \(\beta \) with companions and conjoints, and uniqueness follows from the assumption of an isomorphism.