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 :)