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The study of homomorphisms of $(n,m)$-graphs, that is, adjacency preservingvertex mappings of graphs with $n$ types of arcs and $m$ types of edges wasinitiated by Ne\v{s}et\v{r}il and Raspaud [Journal of Combinatorial Theory,Series B 2000]. Later, some attempts were made to generalize the switchoperation that is popularly used in the study of signed graphs, and study itseffect on the above mentioned homomorphism.  In this article, we too provide a generalization of the switch operation on$(n,m)$-graphs, which to the best of our knowledge, encapsulates all thepreviously known generalizations as special cases. We approach to study thehomomorphism with respect to the switch operation axiomatically. We prove somefundamental results, which, we believe, will be essential tools in the furtherstudy of this topic. We also prove the existence of a categorical product for$(n,m)$-graphs with respect to a particular class of generalized switch. Wealso provide a way to calculate the product explicitly, and prove generalproperties of the product. Also, in the process of proving the fundamentalresults, we have provided yet another solution to an open problem posed byKlostermeyer and MacGillivray [Discrete Mathematics 2004].

Homomorphisms of (n,m)-graphs with respect to generalised switch