The categorical framework of M-adhesive transformation systems does not cover graph transformation with relabelling. Rules that relabel nodes are natural for computing with graphs, however, and are commonly used in graph transformation languages. In this paper, we generalise M-adhesive transformation systems to M, N -adhesive transformation systems, where N is a class of morphisms containing the vertical morphisms in double-pushouts. We show that the category of partially labelled graphs is M,

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... -adhesive, where M and N are the classes of injective and injective, undefinedness-preserving graph morphisms, respectively. We obtain the Local Church-Rosser Theorem and the Parallelism Theorem for graph transformation with relabelling and application conditions as instances of results which we prove at the abstract level of M, N -adhesive systems.