behind materialization: the left-hand side and the edges that are attached to it are “pulled out” of the given abstract graph. [The concrete construction in the category $\GR$ is spelled out in App. \[apx:materialization-graphs\].]{}

We can summarize the result of our constructions in the following proposition:

\[prop:language-materialization\] [ Let $\phi\colon L\to A$ be an arrow in $\C$ and let $L\oset{\eta_\phi}{\rat} {\mat{\phi}} \to A$ be the corresponding materialization. Then we have $$\mathcal{L}(L\oset{\eta_\phi}{\rat} {\mat{\phi}}) =
    \{L\oset{m_L}{\rat} X \mid \exists \psi\colon (X\to A).\ (\phi
    = \psi\circ m_L) \}.$$ ]{}

Characterizing the Language of Rewritable Objects {#sec:matRew}
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match obtained through the materialization of the left-hand side of a production from a given object may not allow a DPO rewriting step because of the gluing condition. We illustrate this problem with an example.

\[ex:POCexample\] Consider the materialization $L \rat {\mat{\phi}} \to A$ from Example \[ex:construction-materialization\] and the pro-\

duction $L~\lat~I~\rat~R$ shown in the diagram to the right. It is easy to see that the pushout complement of morphisms $I \rat L \rat {\mat{\phi}}$ does not exist.\
Nevertheless there exist factorizations $L \rat X \to A$ abstracted by ${\mat{\phi}}$ that could be rewritten using the production.

In order to take the existence of pushout complements 