Too often i come across the statement a functor between two groups as categories is the homomorphism between the corresponding groups. This may be trivial, but has anybody proved is necessary and sufficient condition in detail showing how the object part and arrow part of the finctor can be fixed to prove. Thanks for help
2026-04-30 08:36:59.1777538219
Group homomorphism in category theory
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Well, it is quite trivial: both groups consist of a single object, so the functor can only map the first object to the second. For two arrows $g,h$ in the first group, and a functor $f$ to the second group, functoriality means that $f(gh) = f(g)f(h)$. But this is precisely the definition of a group homomorphism.