Lemma 37.9.5. Same notation and assumptions as in Lemma 37.9.4. There is an action of the sheaf

on the sheaf (37.9.4.1). Moreover, the action is simply transitive for any open $U' \subset X'$ over which the sheaf (37.9.4.1) has a section.

Lemma 37.9.5. Same notation and assumptions as in Lemma 37.9.4. There is an action of the sheaf

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(a^*\Omega _{Y/S}, \mathcal{C}_{X/X'}) \]

on the sheaf (37.9.4.1). Moreover, the action is simply transitive for any open $U' \subset X'$ over which the sheaf (37.9.4.1) has a section.

**Proof.**
This is a combination of Lemmas 37.9.1, 37.9.2, and 37.9.4.
$\square$

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