Lemma 37.9.2. Let $S$ be a scheme. Let $(a, a') : (X \subset X') \to (Y \subset Y')$ be a morphism of first order thickenings over $S$. Let

\[ \theta : a^*\Omega _{Y/S} \to \mathcal{C}_{X/X'} \]

be an $\mathcal{O}_ X$-linear map. Then there exists a unique morphism of pairs $(b, b') : (X \subset X') \to (Y \subset Y')$ such that (1) and (2) of Lemma 37.9.1 hold and the derivation $D$ and $\theta $ are related by Equation (37.9.1.1).

**Proof.**
We simply set $b = a$ and we define $(b')^\sharp $ to be the map

\[ (a')^\sharp + D : a^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'} \]

where $D$ is as in Equation (37.9.1.1). We omit the verification that $(b')^\sharp $ is a map of sheaves of $\mathcal{O}_ S$-algebras and that (1) and (2) of Lemma 37.9.1 hold. Equation (37.9.1.1) holds by construction.
$\square$

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