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I have 4 topoi $A,B,C,D$ (these are associated to abelian sheaves on some sites) and functors (which corresponds to some push-forward of sheaves)

$F: A\rightarrow B$

$G: B \rightarrow C $

$H: A \rightarrow D$

$K: D \rightarrow C$

I know that $G\circ F= K\circ H$.

For every sheaf $a\in A$ (resp. $b\in B$, $ d\in D$) I know how to compute $R^1 H(a)$ (resp. $R^1G(b)$, resp. $R^1K(d)$). Given such information, is there a way to recover

$R^1F(a)$

from these data?

More concretely let $X,Y$ separated, schemes of finite type, $f:X\rightarrow Y$ a projective morphism, $\epsilon_{X}: X_{et}\rightarrow X_{Zar} $, $\epsilon_{Y}: Y_{et}\rightarrow Y_{Zar} $, then $A=D^b(X_{et}), B=D^b(Y_{et}),C=D^b(Y_{Zar}), D=D^b(X_{Zar})$, $a\in Ab(X_{et})$, $F= f_{et,*}, G= \epsilon_{Y,*}, H=\epsilon_{X,*} ,K= f_{Zar,*}$

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A topos is not an abelian category. (The category of internal abelian groups in a topos is.) –  Zhen Lin Jun 14 '12 at 10:24
    
@ZhenLin I edited it –  ulli Jun 14 '12 at 14:31
    
@ulli: Perhaps your question is better suited for mathoverflow. –  Martin Brandenburg Jun 17 '12 at 17:34
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