match $m_L$ is not relevant.

Given a production $p$ and a match $m_L$, if there exist arrows $X \gets C$ and $C \gets I$ that make the left-hand square of the diagram in Def. \[def:dpo-rewriting\] a pushout square, the *gluing condition* is satisfied. If $\C$ is an adhesive category (and thus also if it is a topos [@ls:toposes-adhesive]) and the production consists of monos, then all remaining arrows of double-pushout diagrams of rewriting are monos [@ls:adhesive-journal] and the result of rewriting—be it the object $Y$ or the co-match $m_R$—is unique (up to a canonical isomorphism).

Subobject Classifiers and Partial Map Classifiers of Graphs {#sec:classifiers}
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A standard category for graph rewriting that is also a topos is the category of edge-labelled, directed graphs that we shall use in examples, as recalled in the next definition. Note that due to the generality of the categorical framework, our results also hold for various other forms of graphs, such as node-labelled graphs, hypergraphs, graphs with scopes or graphs with second-order edges.

\[def:category-graph\] Let $\Lambda$ be a fixed set of *edge labels*. A *($\Lambda$-labelled) graph* is a tuple $G = (V_G,E_G,\sSrc_G,\sTgt_G,\sLab_G)$ where $V_G$ is a finite set of *nodes*, $E_G$ is a finite set 