The authors show that the existence of an unbounded and continuous linear operator \(T:\lambda (A)\rightarrow \lambda (C)\) between Köthe sequence spaces \(\lambda (A)\), \(\lambda (C)\) which factors through a third Köthe space \(\lambda (B)\) implies the existence of an unbounded continuous quasidiagonal operator \(Q:\lambda (A)\rightarrow \lambda (C)\) such that \(Q=L\circ R\), where \(R:\lambda (A)\rightarrow \lambda (B)\) and \(L:\lambda (B)\rightarrow \lambda (C)\) are continuous quasidiagonal operators. A continuous and linear operator \(S:\lambda (A)\rightarrow \lambda (B)\) is unbounded if no \(0\)-neighbourhood of \(\lambda (A)\) is mapped into a bounded subset of \(\lambda (B)\). An operator \(D:\lambda (A)\rightarrow \lambda (B)\) is quasidiagonal if there is a sequence \((t_n)\) of scalars and a map \(\sigma: \mathbb N \rightarrow \mathbb N\) such that \(D(e_n)=t_ne_{\sigma (n)}\) for each \(n\in \mathbb N\), where \((e_n)\) is the canonical basis of \(\lambda (A)\).