where the four maps lie in \(L\) and \(R\) as indicated. If the outer square is \(L\)-cartesian, then the bottom square is cartesian.
Proof
By left cancellation, the gap map \(Z \to Y \times_{Y'} Z'\) of the bottom square lies in \(R\). Consider the commutative diagram Then the three maps lie in \(L\) as indicated, hence by right cancellation so does the map \(Z \to Y \times_{Y'} Z'\). Since \((L,R)\) is a factorization system, it follows that this map is an isomorphism, as desired.
