Classifying cofibrations in Mor(M) where M is a model category

Question

If $M$ is a model category the category of morphisms in $M$ has a model category structure where the fibrations and the weak equivalences are the levelwise fibrations and levelwise weak equivalences. I have read that a morphism $f:u\to v$ in $Mor(M)$, which is really a commutative square in $M$ \begin{align*}& \ u_0\overset{f_0}{\rightarrow} v_0\\ & \downarrow u\quad\downarrow v \\ & \ \ u_1\underset{f_1}{\rightarrow} v_1,\end{align*} is a cofibration in this model structure if and only if $f_0$ and $u_1\cup_{u_0}v_0\to v_1$ are cofibrations in $M$. I can prove that this condition is sufficient and I can prove that $f_0$ being a cofibration is necessary but I can't show that $u_1\cup_{u_0}v_0\to v_1$ being a cofibration is necessary. If someone has a nice argument for this and wants to share it or a hint I would be happy=)