In their famous paper "On the differentiation of De Rham cohomology classes..." Katz and Oda construct the spectral sequence for de Rham cohomology for the situation of a smooth morphism

$\pi: X \rightarrow S$

of smooth $k$-schemes ($k$ a field), where $S$ is assumed affine.

There is a step which is not clear to me: in Lemma 8 of this paper they say that the equality

$(*) \qquad \mathbb R^0\Gamma_X =\Gamma_S\circ \mathbb R^0\pi_*$

yields a spectral sequence of composition

$(**) \qquad E_2^{a,b}=R^a \Gamma_S\circ \mathbb R^b\pi_* \Rightarrow \mathbb R^{a+b}\Gamma_X$.

They seem to consider $(*)$ as an equality of functors on the category of complexes of abelian sheaves on $X$ - Arguments involving quasi-coherence and affineness of $S$ only appear in the next step.

My question is a very simple one which I nevertheless can't figure out:

Why do they have $(**)$?

It is a spectral sequence of composition, hence only exists when one knows that $\mathbb R^0 \pi_* $ sends injective objects in the category of complexes of sheaves on $X$ to $\Gamma_S$-acyclic ones. I don't see why this is true: imagine a two-term complex $I^0\rightarrow I^1$ of injective abelian sheaves on $X$, then $\mathbb R^0 \pi_*$ of it is $ker(\pi_*I^0 \rightarrow \pi_*I^1)$. This has no reason to be acyclic for $\Gamma_S$, hasn't it?

A similar problem arises a bit later in Lemma 9 when they consider the equality

$(+) \qquad \mathbb R^0\Gamma_S = H^0\circ \Gamma_S$

which I see as equality of functors on the category of complexes of abelian sheaves on $S$.

Can anybody give a hint how to cleanly resolve these two problems?

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