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== Category-theoretic remarks ==
In [[category theory]], a '''group with operators''' can be defined{{sfn|Mac Lane|1998|p=41}} as an object of a [[functor category]] '''Grp'''<sup>''M''</sup> where ''M'' is a monoid (''i.e.'', a category with one object) and '''Grp''' denotes the [[category of groups]]. This definition is equivalent to the previous one, provided <math>\Omega</math> is a monoid (otherwise we may expand it to include the identity and all compositions).
A [[morphism]] in this category is a [[natural transformation]] between two functors (''i.e.'' two groups with operators sharing same operator ___domain ''M''). Again we recover the definition above of a homomorphism of groups with operators (with ''f'' the [[Natural_Transformation#Definition|component]] of the natural transformation).
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