Lemma 58.3.11. Let $(\mathcal{C}, F)$ and $(\mathcal{C}', F')$ be Galois categories. Let $H : \mathcal{C} \to \mathcal{C}'$ be an exact functor. There exists an isomorphism $t : F' \circ H \to F$. The choice of $t$ determines a continuous homomorphism $h : G' = \text{Aut}(F') \to \text{Aut}(F) = G$ and a $2$-commutative diagram

The map $h$ is independent of $t$ up to an inner automorphism of $G$. Conversely, given a continuous homomorphism $h : G' \to G$ there is an exact functor $H : \mathcal{C} \to \mathcal{C}'$ and an isomorphism $t$ recovering $h$ as above.

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