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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).