Let \((L,R)\) be a factorization system on \(C\). Any morphism \(f\colon X \to Y\) which is both in \(L\) and in \(R\) is an isomorphism.
Proof
Consider the commutative square: Since \(f \in L\) as the left vertical map and \(f \in R\) as the right vertical map, orthogonality provides a unique filler \(g\). The two triangles say precisely that \(gf=\id_X\) and \(fg=\id_Y\), so \(g\) is an inverse to \(f\).
