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 $A=D^b(X_{et}), B=D^b(Y_{et}),C=D^b(Y_{Zar}), D=D^b(X_{Zar})$, $a\in Ab(X_{et})$, $F= f_{et,*}, G= \epsilon_{Y,*}, H=\epsilon_{X,*} ,K= f_{Zar,*}$