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Could someone tell me how the automorphism group of a functor is defined? I tried to google it for quite some time but nothing useful came up.

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In any category, the automorphism group of an object is just the set of isomorphisms from that object to itself, with the group operation being composition. A functor $F:\mathcal{C}\to\mathcal{D}$ is an object of the category $\mathcal{D}^{\mathcal{C}}$, where the isomorphisms are natural isomorphisms of functors, so the automorphism group is the set of all natural isomorphisms $F\to F$ under composition.

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