The Stacks project

Lemma 58.4.5. In diagram (58.4.0.1) the following are equivalent

  1. the image of $h'$ is normal, and

  2. for every connected object $X'$ of $\mathcal{C}'$ such that there is a morphism from the final object of $\mathcal{C}''$ to $H'(X')$ we have that $H'(X')$ is isomorphic to a finite coproduct of final objects.

Proof. This translates into the following statement for the continuous group homomorphism $h' : G'' \to G'$: the image of $h'$ is normal if and only if every open subgroup $U' \subset G'$ which contains $h'(G'')$ also contains every conjugate of $h'(G'')$. The result follows easily from this; some details omitted. $\square$

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