Still trying to figure that out. It looks like you start with a partial function L->R instead of a complete function, so some of the vertices or edges in L don't map to anything. You can decompose that into two morphisms L <- G -> R where the left map is injective. So G identifies the elements to keep, L is a superset of that which gives you the "match" including some elements to delete, and R is the result.

I believe the two pushouts are constructed on these two morphisms and two of something else, but that's the point where Wikipedia is vague and the article I linked to is complicated, so I'm a bit confused yet.