Why does this construction give a weak factorization system in the category of span diagrams? 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.
 A: 
they seem to implicitly use the fact that (in their notation) if a map f is such that fa, fb, and fc are acyclic cofibrations, then ia(f) and ic(f) are again acyclic cofibrations.

No, that's not what they're using.  They're establishing the lifting property
of acyclic cofibrations with respect to fibrations.
Acyclic cofibrations are by definition cofibrations that are weak equivalences.
They define cofibrations as maps $f$ such that $i_a(f)$, $i_b(f)$, and $i_c(f)$
are cofibrations in $C$.
Weak equivalences are defined indexwise.
Thus, acyclic cofibrations are precisely thos maps $f$ for which
$i_a(f)$, $i_b(f)$, and $i_c(f)$ are acyclic cofibrations in $C$,
as follows from the 2-out-of-3 property for weak equivalences.
This model structure is a special case of the model structure
on directed diagrams (such as $C^D$),
which itself is a special case of a Reedy model structure.
In this case, both nonindentity arrows in $D$ are positive.
The (acyclic) cofibrations are precisely
Reedy (acyclic) cofibrations in the Reedy model structure,
and the functors $i_a$, $i_b$, $i_c$ are precisely the Reedy latching maps.
