This I followed along okay, which surprised me because I generally stay away from category theory, and that initial Wikipedia link looks like it is written in language. I like the diagrams as well, they were very helpful.
The removal process, does it depend on finding a sort of inverse for a morphism and a pushout?
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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.