The study of homomorphisms of $(n,m)$-graphs, that is, adjacency preserving vertex mappings of graphs with $n$ types of arcs and $m$ types of edges was initiated 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 switch operation that is popularly used in the study of signed graphs, and study its effect 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 the previously known generalizations as special cases. We approach to study the homomorphism with respect to the switch operation axiomatically. We prove some fundamental results, which, we believe, will be essential tools in the further study 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. We also provide a way to calculate the product explicitly, and prove general properties of the product. Also, in the process of proving the fundamental results, we have provided yet another solution to an open problem posed by Klostermeyer and MacGillivray [Discrete Mathematics 2004].

翻译：对$(n,m)$-图（即具有$n$类弧和$m$类边且保持邻接关系的顶点映射）的同态研究由Nešetřil和Raspaud发起[Journal of Combinatorial Theory, Series B 2000]。此后，学者们尝试推广符号图研究中广泛使用的切换操作，并考察该操作对上述同态的影响。本文同样提出$(n,m)$-图上切换操作的一种推广形式——据我们所知，该推广将此前已知的所有推广形式作为特例加以涵盖。我们采用公理化方法研究关于切换操作的同态，证明了一些基础性结论，相信这些结论将成为该主题后续研究的重要工具。我们还证明了在特定广义切换类下$(n,m)$-图范畴积的存在性，给出了该积的显式计算方法，并证明了积的一般性质。此外，在证明基础结论的过程中，我们为Klostermeyer和MacGillivray提出的一个公开问题[Discrete Mathematics 2004]提供了新的解答。