Lemma 57.4.2. In diagram (57.4.0.1) the following are equivalent

1. $h \circ h'$ is trivial, and

2. the image of $H' \circ H$ consists of objects isomorphic to finite coproducts of final objects.

Proof. We may replace $H$ and $H'$ by the canonical functors $\textit{Finite-}G\textit{-Sets} \to \textit{Finite-}G'\textit{-Sets} \to \textit{Finite-}G''\textit{-Sets}$ determined by $h$ and $h'$. Then we are saying that the action of $G''$ on every $G$-set is trivial if and only if the homomorphism $G'' \to G$ is trivial. This is clear. $\square$

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