Exercise 1.3.30


Consider and an arbitrary group as one-object categories. For each , there is a unique functor that sends the arrow to the arrow (such a functor is really just a group homomorphism). Natural isomorphism between such functors is an equivalence relation. Suppose and are naturally isomorphic, say via (note any natural transformation in this setting is necessarily a natural isomorphism, since all arrows are isomorphisms in groups). only has one component, so we will just abuse notation and refer to the component as , which is an arrow/element of . The only thing that “matters” then, about is the naturality relation, which simplifies to the following: For any an arrow/element of , we have . When , we have , i.e. . So, natural isomophism of and implies there exists an such that this occurs – which is to say that and are conjugate. So, equivalence classes of these functors under natural isomorphism is the same as conjugacy classes of the group elements.