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I'm doing a little homework but I think my brain has ceased to function late at night, was hoping you could help me out with what I am certain is a very simple group/cat theory problem. Similarly to in a group, we define an inner automorphism of a category $\mathcal{C}$ to be an autmorphism with a natural isomorphism to the identity functor on $\mathcal{C}$ (where an automorphism is a functor $\mathcal{C} \to \mathcal{C}$ with 2-sided inverse, and a natural isomorphism is a natural transformation with all components invertible).

All I want is to show that the inner automorphisms form a normal subgroup in the group of all $\mathcal{C}$-automorphisms, i.e. for any inner automorphism $i$, $gig^{-1}$ is an inner automorphism for any automorphism $g$. Presumably I just want to take the natural isomorphism which we have for i, then modify it very slightly to something new which works for $gig^{-1}$.

I know I did this for groups years ago and am fairly certain it should be painfully simple but on a lack of sleep I don't seem to be getting anywhere with it! Any help you could offer would be much appreciated, thank you :)

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apply $g(-)g^{-1}$ to the natural transformation to the identity functor. Applying it to the identity functor should give the identity functor, and applying it to a natural transformation should give a natural transformation. –  Jack Schmidt Oct 24 '11 at 21:12
    
Ah, I knew it was simple! Being stupid. Thanks Jack. –  Spyam Oct 24 '11 at 21:40
    
@JackSchmidt Since your comment appears to have answered the question, could you promote it to one (so as to remove this -- effectively already answered -- question from the unanswered queue)? –  Lord_Farin May 23 '13 at 11:55

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