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The following result seems to be well known, but I can't come up with a proof.

Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is any abelian sheaf on $C$, then the object $RHom(G,P)$ is computed by the bicomplex $$ RHom(F,P)\to RHom(F\times_G F,P)\to RHom(F\times_G F\times_G F, P)\to \dots $$

Any ideas? Thanks!

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See SGA4.V.7, especially Theorem 7.4.1. – Jonathan Wise Jan 29 '13 at 20:58

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