Why does this construction give a weak factorization system in the category of span diagrams?

Question

In Dwyer and Spalinski's classic paper Homotopy Theories and Model Categories, they describe homotopy pushouts by defining a model structure on the category of span diagrams in a given model category $C$. In their Prop 10.6, in proving this is a model category, specifically in proving the weak factorization axiom, they seem to implicitly use the fact that (in their notation) if a map $f$ is such that $f_a$, $f_b$, and $f_c$ are acyclic cofibrations, then $i_a(f)$ and $i_c(f)$ are again acyclic cofibrations. However, it is far from clear to me why that ought to be the case. It is easy to see that they are weak equivalences using 2-out-of-3, but I can't seem to find a reason they should be cofibrations.